It looks like there is a newer, fresher-looking format of the Titan status indicator. It seems to currently show that the titan is up and holding its position (starting at 0:13), as symbolized by the yellow shield symbol, much like you could tell your titan to hold position or follow you in the first Titanfall game. At 0:18, the symbol behind the titan changes to a green double-arrow symbol, possibly representing the "follow you" command, as the titan comes closer to the player as she grapples over the enemy titan.
Next, you can see the L1 and L2 buttons in the bottom-left, which are the Playstation versions of LB and RB on Xbox. These same buttons seem to correspond with tactical abilities (Active Radar Pulse, grappling hook, Stim, Cloak, etc) with the L1/LB button, and ordinance (Arc Grenades, Frag Grenades, Arc Mines, Satchel Charges, etc.) with the R1/RB button.
The last indicator here is a simple weapons indicator, which is self-explanatory.
The map doesn't seem complicated.

Over the course of the trailer, and the switching between different players' point of views, you will see many different symbols for the Tactical Abilities, Ordinance, and for the Weapons (which I will get to later):

0:13-0:23 has a grappling hook symbol for the tactical ability and a lightning-bolt symbol in a cloud shape for the ordinance. The grappling hook seems to have two charges, and when the player activates it at 0:20, it starts a cooldown time on the grappling hook icon. So far, you can use the grapple to hook onto Titans (0:20), walls/structures (1:03), AND other players ([urlhttps://youtu.be/2ISYN1AshZY?t=1m37s]1:38[/url]). The ordinance could be an Arc Grenade.
[urlhttps://youtu.be/2ISYN1AshZY?t=28s]0:29-0:32[/url] has a pointer pointing at some pulse waves for the tactical ability and a bag with a handle for the ordinance. Both the tactical ability and ordinance are featured in the clip running from 0:59-1:01, as the L1 ability appears to be a stationary radar pulse grenade, while the ordinance is the Satchel Charges.
0:31-0:34 has a syringe in front of a green arrow (probably the stim ability) in L1, and a shuriken shape in front of a blue arrow pointing down for the ordinance. The ordinance is a void grenade that works much like the "Void's Tear" Plasma Pistol does in Halo 5, and is featured in the clips running 0:49-0:53.
0:38-0:46 has a shuriken shape in front of an orange flame symbol, which could be an incineration grenade.

The Titan HUD still includes the Core ability in the Titan status ring, the new "Defensive" Ability in L1, the Tactical Ability with the Playstation "Δ" symbol and the Ordinance in R1. During the Titan clips, there is the clip going 0:34-0:36 that has an extra symbol above the L1 ability. This extra symbol has the control prompt of pressing left on the D-pad to activate it. I don't know what that is either.

Same as with the Pilot gameplay, over the course of the trailer, and the switching between different players' point of views, you will see many different symbols for the Core Abilities, Tactical Abilities, Ordinance, the new "Defensive" Abilities, and Weapons (which I will still get to later):

As detailed here (the Titanfall website), there are 6 titans going to be in the multiplayer, so far. 3 of them are revealed, with only a couple actually previewed in this trailer.
0:27-0:29 has a "Flame Core" Ability, a flame in front of a shield for the "Thermal Shield" Defensive ability, a flame in a diamond-shape for "Firewall" Ordinance, and a spinning gas canister in a green triangle-shape for the "Incendiary Trap" Tactical ability. This is the loadout of the "Scorch Titan". The Core Ability is described here, as "creates a giant wave of fire that destroys anything in its path" that deals immense amounts of damage and leaves a trail of fire in its wake. This ability is also featured in the clips running 0:31-0:34, 1:09-1:22, and 1:26-1:40. The tactical ability is an incendiary trap, describe by the official Titanfall page as an ability that "deploys flammable gas canisters which can be triggered by the slightest spark." It is a green triangle-shaped symbol with a small gas canister with waves of gas spiraling out of it. The ordinance ability is featured in 0:56-0:59, and is described on the official Titanfall page as an ability that "fires a directed wall of thermite in the enemy's direction". The Defensive ability is not preview in this trailer, but is described (on the Titanfall page) as a shield that "liquifies incoming artillery while dealing damage to nearby enemies."
1:26-1:40 has a "Laser Core" Ability, a blue spiral "Vortex Shield" for the Defensive ability, a targeting "Laser Shot" for the Ordinance, and a "Tripwire" mine in a red triangle for the Tactical ability. This is the loadout of the "Ion Titan", as described by the official Titanfall webpage here. The Core ability is featured in this same clip as well as 0:54-0:55, and seems to be a super-discharge of the core, focused into a single beam to damage the target. It's described on the Titanfall webpage as "a deadly oversized laser beam fired from the chest of the Ion and capable of stopping anything in its path." The spiral vortex defensive ability seems to be very similar to the Vortex Shield that was in the original Titanfall. It's also described as a shield that "absorbs incoming fire and redirect it at the enemy", and can be seen in the clip running 0:27-0:29 in the background. The Laser-Shot ordinance small anti-infantry device that destroys small infantry or pilots, or maybe even incoming ordinance. It shows up, listed as a "Laser Shot" in the kill-feed, and is described on the Titanfall page as a "shoulder-mounted... lethal beam that cuts through anything in its way". This ability/equipment can also be seen on the right shoulder of the titan the player jumps over in the second clip in the trailer, at 0:19-0:23. The Tactical ability is not shown in the trailer, but is described as a "Laser triggered explosive mine", and symbolized by a mine on a blue triangle.
NOTE: As the clips I could make out were either all fire-themed or all laser-themed (pertaining to the Scorch and Ion titans, respectively), these are probably based on the different Titan model you pick. This is similar to how you would get the Speed Core with the Stryder, Damage Core with the Atlas, or the Shield Core with the Ogre Titan in the original Titanfall.
There is one Titan that is not previewed at all, but is described on the official Titanfall page here. It is the "Ronin" Titan, and has a "Sword Core," swords in front of a shield for the "Sword Block" Defensive ability, a lightning bolt in front of a blue diamond for the "Arc Wave" Ordinance, and an arrow in front of a green triangle for the "Phase Dash" Tactical ability. The Sword Core is described as a core that "empowers his melee and sword abilities, and gives access to new sword attacks". This has me assuming that the sword is the default weapon for this titan, and that normally, there are simple attacks, but the core unleashes faster, more dangerous attacks. The Sword Block is described as a shield that "reduces damage taken... can be held indefinitely", which is great to be able to have this ability at any time, rather than have to wait for a cooldown. The Arc Wave's description is "swiping his sword across the ground, Ronin creates an electrical wave that damages and slows enemies hit by it". This sounds like something that is not viable for anti-air combat (mid-air pilots, for instance), but it could have an electrical wave that goes outward and upward that would help with that. The Phase Dash seems fantastic, as it's described "Ronin dashes in a given direction, temporarily phasing out of the world, avoid incoming fire and creating the element of surprise". The Ronin must have its own personal warp transmitter, letting it cover some distance. I wonder if this would also allow large distances to be covered quickly, such as cross-map, with a good view?

0:02-0:10 In this clip, the person to the left of the player seems to be checking a Carbine-type weapon at the beginning of the clip. When she is jumping out, she has a weapon similar to the Charge Rifle on her back. The person in front of the player seems to be wielding a LMG, based on the fact that it has a carrying handle on the front of it.
0:13-0:23 For this section, the player seems to be wielding a Carbine or SMG that holds 40 rounds in a magazine or clip. The secondary weapon (as represented by the dark silhouette under the weapon marker), seems to be a regular handgun, but it could be semi-automatic or automatic. In the kill-feed, there is a "T-203 Thermite Launcher" listed, which I'm assuming is a Titan weapon much like the Triple Threat. I can only speculate, but it looks like the red Titan the player jumps over may be equipped with the Quad Rocket Launcher, but it is hard to make out. Same with the rest of the clips. I am not able to make out any details on any weapons, unfortunately.
0:29-0:31 In this clip, the player uses a 4-round shotgun to obliterate an opponent, which is labeled as the "Mastiff" in the kill-feed. The secondary weapon appears to be a rifle of some sort. Also at the beginning of this clip, it seems that Archer missiles are making a comeback, based on the kill-feed.
0:31-0:34 Here, the player is wielding a R201 Carbine, which has replaced the R-101C Carbine from the original Titanfall. The player switches to the secondary weapon at the very end of the clip, and it looks a lot like the Archer missile launcher from the original Titanfall. Other weapons that appear in the kill-feed are the Hemlok (which was the Hemlok BF-R in Titanfall 1), CAR (which was the C.A.R. SMG in Titanfall 1), Thermite Meteor, and the Softball. The "Thermite Meteor" may be the fire-in-diamond-shape Titan ordinance that was mentioned in the section above.
0:37-0:46 For this clip, the player is wielding the new CAR SMG (as represented by the kill-feed), which may have at least a 36-round clip or magazine. When he first walks through the wall opening, there is a chance for him to pick up an "Alternator" (with Extended Mags and Speedloader), which is explained more in the next clip. The sidearm still looks like the basic sidearm silhouette we saw in the second clip of the trailer. In the kill-feed, the only new weapon that shows up is the Kraber, which could be the new version of the Kraber-AP Sniper from the original Titanfall.
0:49-0:54 The player we are viewing is wielding the Alternator, which appears to be an SMG with a little wider spread of fire or faster firing rate. This is because it alternates firing from to barrels, lef-right-left-right-left-etc, as you can see by the action bolts on the weapon as it is firing. It also seems to have a 25-round capacity after being reloaded. This is labeled more specifically as a "Twin Barrel SMG" in the clip running from 1:26-1:40.
1:00-1:01 The only new weapon we see here is the "L-Star", as it is listed in the kill-feed.
1:02-1:08 This is where we get to see the L-Star in action, as it seems to be a lower-damage SMG-style weapon with unlimited ammo. Perhaps it overheats or has to discharge? There is a "Mozambique" weapon listed in the kill-feed as well.
1:09-1:22 For this clip, the only new weapon we see in this clip is the "Double Take", as it is listed in the kill-feed. This could be a semi-auto sniper rifle, much like the Longbow-DMR Sniper.
1:26-1:40 The titan here is wielding a weapons listed as follows: "Splitter Rifle - Drains energy to split the beam, increasing damage." It seems to work much like the Sidewinder of the original Titanfall, only bigger. It seems to have a 40-round capacity, when reloaded.
Other "weapons" that appear in the kill-feed that are self-explanatory: "Titan Punch", "Satchel", and "Melee".
NOTE: So far, it appears that the R-101C Carbine (replaced by the R-201 Carbine), EVA-8 Shotgun, Smart Pistol MK5, R-97 Compact SMG, Longbow-DMR Sniper, G2A4 Rifle, Spitfire LMG, RE-45 Autopistol, Hammond P2011, and the B3 Wingman, all are not returning from the original Titanfall. That is what I could find based on the June 12th, 2016 Titanfall 2 Gameplay Trailer and this source (although that source isn't the most... accurate).

- It seems that there are different models of Titans. I count 2 sets of abilities, but the clip running from 0:13-0:23 kind of makes it look like there may by at least 4 different types (because all of them looked a little different, which they could look different because of different equipment).
- I didn't see anything about a Cloak, or cloaking device. I really enjoyed using that in Titanfall 1 and hope it doesn't get left out.
- Some of the titans are labeled as "Enhanced Auto-Titan", while others are just labeled "Auto-Titan". This is shown with two ally Titans with those different titles at the beginning of the clip running 0:13-0:23.
- In the kill-feed, it will now show up as separate lines for killing a Pilot, and killing a Pilot's Titan. For example: "Fundo [Flame Core] Swift's Titan"; "Fundo [Flame Core] Swift", as evident by the clip running 0:27-0:28.
- There seems to be some sort of Pilot ability that makes a hologram version of yourself, which is shown(?) in the clip running 0:37-0:49.
- A new ability could be hovering in midair, like you can do when you smart-link mid-jump in Halo 5. I'm not sure if it's related to the L-Star, but it is seen in 1:05-1:08.
- For filling up the meter to get your Titan into play, it seems that it will be easier for skilled players. From what I can see, when something contributes to the meter, it contributes a blue section and an orange section. The Blue section is permanent and does not decrease, but it looks like the orange section slowly decreases, as shown in most clips, but especially 0:37-0:46.
- Lastly, there seems to be a special ability unlocked by filling up your meter about two-thirds of the way. You can see the marker and flame symbol for it first in the clip running 0:29-0:31, and you can see someone earn the special thing in the clip running 0:49-0:53.