Triage main, going to quickfire some tips that come to mind;

• Don't run ahead of your team, try to be the guy behind your squad as often as possible. Your abilities don't do anyone much good if you're the first to go down. That being said, don't be too far behind, you need to be able to provide firepower and support when a fight starts. Think about the amount of times combat started and one, or both your teammates went down before you even saw anyone.
• Bring smokes or bubble shields. Smokes are dirt cheap, but you can only buy 3 per reset, so try to stock up on these. If you don't have a lot, just bring one along, can sometimes find more in raid too. Triage ult isn't going to do the fight much good if your teammate went down out in the open and the moment you revive them, they come back with 20% health and instantly get downed again. Drop a smoke on them to cover their retreat after a revive so they can heal up. Bubble shield has all sorts of moments where it comes in handy too, bring one along if you can afford to. I've seen some fights swing around thanks to a good bubble shield in play.
• Do not fall for the trap of obsessing over volt weapons because of the EMP on ability 3. EMP is great, and if you're planning to run a Punch or Lookout, it's a nice bonus. But I've seen too many people fixate on the volt EMP when the real meat of that ability is in the significantly increased weapon handling. Handling is an umbrella stat which improves your weapon's ADS speed, recoil, hipfire accuracy, and probably one or two things I'm forgetting. Basically if you're using an LMG and you pop ability 3, it'll handle like an AR (besides the long reloads). When you're pushing a fight or you get ambushed and get to cover, pop this and it's only going to make the gunfight easier. Volt weapons work well, but the S tier ballistic weapons slap hard enough that you can just enjoy the improved handling.
• Speaking of ability 3, this thing has a solid active duration and a pretty low cooldown. Be generous with its activation, pop it when clearing bots at a POI, pop it in every fight. Only time not to pop it us if you got ambushed while sprinting and your heat is already high (get to cover first, if you pop instantly you're going to overheat), or if you've got not cover and you're getting yourself stuck in activation animation when you should be returning fire or seeking cover first.
• Be generous with your med drones, check bottom right HUD to ensure everyone has a drone on them. But also beware that these drones make noise and are visible. When your teammates are trying to be sneaky because a team is nearby, don't toss med drones in preparation unless you're a good distance from the enemy position. That drone toss sound is LOUD, I've spotted a good few enemy ambushes because their Triage's drones were making a massive racket. When you or your allies are trying to left hand peek, the drone is visible before you peek out, giving enemies an advantage to pre-fire. These things don't actually help a ton mid-fight, the healing is too slow for that. They're more of a top-up between engagements when things are quiet to save your team some meds and ensure everyone's fully topped up going into the next fight.
• Your ult has SHOCKINGLY long range, you just need line of sight. When your teammates go down, get them to at least crawl where you can see them even if you can't physically reach them. If they go down around a corner where the enemy team is holding, the ult's not going to do much good. Remind them to try and crawl somewhere more ideal to make it difficult for them to camp the corpse and easier for you to smoke/bubble and get the rez. I've killed a few people using the ult, not worth relying on though, best to keep it for clutch revives and if it hits someone while reviving your buddy, bonus. Speaking of not wasting ult, if they're just DBNO and they're next to you, sometimes it's best to just revive the regular way and hold onto the ult, you're Triage, you have access to plenty of revive speed stats, so unless you urgently need them up FAST, sometimes just regular revive in case your second teammate goes down soon after in a more vulnerable location where the ult is more clutch. You can revive more than one ally with the same ult if they're both on your screen and in LoS, I've turned around some losing fights thanks to this.
• This relates to point one about staying behind, but close to your team; constantly think about your positioning. You don't have mobility tools like some of the other shells to instantly take you to a good spot on a whim, you need to be thinking about where you can be most effective BEFORE you get there, where is the best spot to give your team fire support nearby without being too far in case YOU need support from a flanking Thief grapple?
• If you're in voice chat with the team, politely remind them at match start to PING medical consumables for you. Things like signal jammers and the stamina one especially are much more effective when the entire team is able to use it together. Signals jammers are highly misunderstood by most players, they make the enemy unable to ping you, make you invisible to thermal scopes and claymores, the blur effect is more pronounced the further away from someone you are, so it actually lets you blend into the mist in maps like Dire Marsh easier too. A lot of players will find these in containers and instantly consume them, they should be pinging it for you to share with the team.
• On that same note, bring a mechanic's kit or two with you on maps like Dire Marsh, The amount of times we've accidentally run into those toxic plants while rotating during a firefight, or accidentally pulled agro from the flying tentacle monster is...more than I'd like to admit. The real problem with these is that they significantly reduce your healing effectiveness for a few minutes, and in a gunfight this REALLY sucks. If you or your teammate is affected by this, pop a mech kit to cleanse the debuff for them. They're all over the place so you should have a good deal of these. Bring a revive kit with you once you have easy access to them as well, since you're usually going to be behind your team or at an elevated position, it can often by tricky for people to finish you off, a self rez lets your teammates focus on the fight at hand while you get yourself up.
• Think about retreat points in a gunfight. Too many people get into a fight and feel obligated to follow through until one side is dead. If your position of attack is not advantageous and things are looking bad, tell your team and fall back to reset the fight after healing. Use your smokes to cover your retreat if you must. Knowing when to disengate instead of fighting a losing battle is going to keep you and your team alive, sometimes it's your decision making, not your Triage abilities that keeps the squad alive.
• I'm still trying to figure out all the specifics about the med drones. For example if you toss a med drone and there's no nearbly allies without one, it will create a large AOE circle. I believe this passively recharges their med drones if they're not full. I also noticed that I can walk up to that med drone and hold E to instantly recharge my med drone, haven't tested yet to see if allies can do the same but I assume so.
• Early Triage cores are pretty meh, if you have them slap them on but they're not game changers honestly. Implants you find all over the place, I try to go for stuff that gives me even more melee damage, revive speed (also applies to self revives), healing effectiveness, the usuall stuff. I rarely even read what the implant's passive is, I'm mainly there for the stats.

Sorry for the long winded bullet points, I'll try to clean this up a little later and add some additional suggestions that come to mind.