My thoughts on a few things (I don't have experience enough around ones I didn't comment on):
P-2 Peacemaker - Since it's your only pistol, fully upgrade it for the times you may be stuck using it. It's decently powerful, but I wish I could switch it for other one-handers.
AR-Liberator - Decent starting weapon, but becomes less useful starting at around difficulty 5 I'd guess.
RX1-Rail Gun - This thing is awesome, but only holds 8 clips and less rounds per clip than the LHO-63 Camper. The big bonus of this is that it goes through a LOT of enemies and can stun tanks, which makes it insanely useful to have at least one in the group using it.
LAS-5 'Scythe' - This is a good gun that starts to lose its usefulness at around stage 7 or 8. Enemies start taking too long to kill, but before that it's not bad. It's particularly good at sweeping the smaller enemies very quickly and taking them out before they can call reinforcements. Cyborgs are a little different though as their reinforces are a bit more durable.
LHO-63 'Camper' - A very good gun, but early on it's a bit of a pain to manage. It takes out larger enemies with 1 bullet and later difficulties you'll need two. However once the Rail Gun is unlocked, there's not much more reason to use this.
Resupply - Pretty self explanatory and useful to have at least one person in the group with it, potentially two.
RL-112 Recoilless Rifle - Great against tanks, especially once fully upgraded. However the pack that drops with it replaces a resupply pack if you're wearing one at the time of pickup. Only 7 total shots, so it's a little less useful later one. Users with the rocket pack have to actually resupply the rocket guy after they fire, as opposed to the way Resupply Packs work.
EAT-17 - The big thing over the RL-112 this has is that you can call it once every 2 minutes. So it has some pros and cons in comparison because of that.
Resupply Pack - Drops a pack that anybody can pickup and players can resupply their ammo by going up to the guy with the pack and hitting X. Alternatively if the guy with the pack is on his last magazine, he can resupply himself. Picking up ammo boxes will add those to the resupply pack as well, which is pretty useful. Cyborg areas tend to have a lot more ammo laying around.
REP-80 - The 'healing gun'. It can heal vehicles and people. Upgraded it's quite nice. It doesn't hurt to have one person using this to help keep mechs or vehicles alive as well as any nearby allies. Only downside is you can only call it in once per mission, so plan accordingly.
EXO-44 'Walker' Exosuit - This thing is great for normal enemies and absolutely tears them a part while holding a TON of machine gun ammo. The downside is you only ever have 8 rockets that can be upgraded to anti-tank rockets with no way (to my knowledge) to refill them.
M5 APC - Great for traversing snowy and swampy terrain. However drivers can easily kill you if you aren't careful. The side guns work quite nicely. As for the main turret, it's alright but only really decent once it's upgraded to laser sight for it.
A/MG-II Minigun Turret - This is a great stratagem, but in later difficulties it will prove a lot less useful due to how frail it is. You can go prone and to prevent getting hit by bullets, which makes earlier difficulties a breeze. However once you start going up against tanks, that's when their usefulness starts to drop, but not entirely until a couple difficulties after.
'Humblebee' UAV drone - Really a must have if you're farming samples and only one person needs it. Expands radar detection range, shows where hidden objectives are (Places in the ? places) and can display samples from a large radius. Everyone gets the benefit from one person summoning the drone.
Static Field Conductors - I thought this would be less useful initially, but my god they're great. It slows practically everything, including allies. Can make defending an area so much easier. I haven't run into anything that it doesn't slow yet, but I haven't tried it on any 'Let's check out difficulty 12' runs.
Reinforce - Upgrade it. Don't not upgrade it. Do it.
NUX-223 Hellbomb - When you first start running into targets it works well enough. However at higher difficulties, please upgrade this. The increased durability and arrival time make such a huge difference. Unless a tank comes in at the last second.

tanks.
Laser Aim Module - Really useful if you're using a the LHO-63 'Camper' or Rail Gun. Otherwise I'd probably choose something else.
MD-99 Autoinjector - This can literally be the difference between living and dying. You regen your health incredibly fast and can get yourself up extremely fast. It won't save you against a group of enemies beating you at the same time, but you can pull yourself back up against single hard hitting enemies before they take you out.
Cardio Accelerator - You always run at full speed. Great for farming Samples for research.
Stun Grenades - This can be useful when you're getting overwhelmed, but if someone has the Rail Gun their usefulness diminishes.
Strategem Priority - If you're using a lot of unlimited use stratagems that have cooldowns before they can used again, this can be insanely useful. Otherwise it's not really worth it. An example of a stratagem for this to be useful would be the Static Field one that slows enemies which move through it or the EAT-17 Rocket Launcher.
A couple other tips:
- Don't be afraid to die. When you come back, you get your ammo back (unless you had more than half your clips).
- See what other stratagems others in your group are using and adjust accordingly.
- It's always useful to have at least one person with an anti-tank weapon as well as one with a REP-80 healing gun.
- Don't bother with samples on higher difficulties, just farm them on lower ones.
- Engaging the enemy isn't the objective except during kill missions (which are easily achieved by just trying to ignore everything)
- Don't be afraid to run away from groups of enemies.
- On higher difficulties, go for the harder objectives first. These include capturing points simply because you need to have everything dead in the vicinity before the bar moves and if you leave it as the last one it can make things
incredibly difficult. Last ones attempted should be escorts or defending objects because failing them is better than failing the overall mission.