Always tail the host unless stated otherwise.
Do not accept any side missions unless stated otherwise.
If Claptrap asks for a high five or Aurelia presents a contract, you melee it.
If someone is down, gauge whether it would be better to A) weaken a nearby enemy B) Freeze an enemy C) quickly clear the area or D) manually revive with some combination of A or B.
Revive with oxygen. ALWAYS.
If you open a chest and it has ammo, press whatever key you have bound to pick items up and grab it all at once as the ammo benefits everyone.
EDIT: added stuff
Athena with “Clear”. You may still want to manually revive. Reason being, you’re still mostly invincible while the shield is up and you can store damage while reviving. Plus, if you started reviving right after activating the shield, you can still put yourself in between the enemy and your buddy so your buddy doesn’t die again immediately. Also, if you manually revive one person but a second person is down but farther away, you can throw the aspis and revive the second person while still getting the first. Lastly, if you have points in Vanguard, the extra health regen will always be appreciated.
Claptrap: Avoid negative action packages. Clap in a box = not good, Torgue Fiesta is questionable but OK as long as you stay away from everyone.