So, if you want to powerlevel your multiplayer Vanguard but don't like the risk inherent with Biotic Charging in and then Nova'ing your shields to oblivion, try the Asari Vangaurd! Stasis bubble trap + sniper rifles = hahahaha holy shit how am I already level 20
Same goes for Asari Adept if you actually manage to unlock it. The Premium Spectre pack has a really good chance for it; granted I'd already unlocked Asari Adept, when I managed to get my first Premium Spectre pack it had another Asari Adept in it (as well as a Salarian Infiltrator card).
Human Sentinels can play like tankier Biotics. Just max out Throw and use it with Tech Armor left off and you can spam it fast enough to kill basic mooks in a damned hurry, and it staggers anything with shields. Viable tactic all the way through Silver challenge. Turn on Tech Armor if things get dicey and you have to sprint away from wherever you're fighting from, or if you're doing one of the objective waves that involves staying in one spot (four bits of tech or data upload/download).
Vanguard players; no, you are not being cute by Charging in and slaughtering things that the Krogan is trying to melee. Krogan players get massive defense boosts and melee damage buffs added when they kill things with melee, but they have to KEEP killing via melee to keep that buff going. Otherwise they are deceptively fragile. Don't deny a Krogan the kills he needs, but at the same time, don't leave him high and dry.
Asari classes don't have a combat roll per se as they have a short-range biotic charge. This eats your shields a little but with proper timing you can dash forwards through missiles. Don't bother trying that unless you're the host, though.
If you don't mind it, you can completely max out all but one entire skill if you don't put any points whatsoever in it. Don't feel like using Inferno Grenades on your Krogan Soldier? Then you can max out Krogan Battlemaster, Rage, Carnage, and Fortification completely. This is best done with the human classes, because you get two of the same skill tree because of the gender options. Make a slightly more balanced build on Male, and a specialized build on Female if you want.
A lot of Biotic skills combo together. Warp is usually used to detonate skills - everyone knows Singularity and Warp, but if you Warp something first then hit it with Throw, you'll set off a detonation, as well. Not sure about Warp + Pull, but Reave and Warp in conjunction do the same, as well as applying heavy defense debuffs.
Tech skills can combo, too. The timing window for setting off a Tech Burst is much shorter than a Biotic Explosion, though. This is best done by working in tandem with another Tech wielding player. Some Biotic skills can trigger a Tech Burst, as well - Biotic Charge after an Overload, for example. Incinerate and Concussive Shot is one such combo, if I'm not mistaken. I could be - I focus mostly on Biotics over Tech.
If you happen to get the N7 pistol, despite being able to upgrade it easily (if at all) it is one of the best lightweight guns in the game. It eats through shields and armor just as quickly as an SMG or Assault Rifle would shields, or a sniper rifle or shotgun would armor. It eats ammo quickly, though. The Predator is usually a better resort; good ammo capacity, excellent damage, and just as effective against armor as it is health.
The Mattock is the better choice for basic Assault Rifles, if you don't have the N7 AR or one of the fancier, high-end ones like the Turian rifle. The Mattock is a single-shot rifle that is incredibly effective against armor as well as health.
Despite my class, the Mattock or the Predator are the only weapons I'll ever use - and only one at a time, no less. If you're playing a biotic-centric class, don't pack two weapons. Use your powers more than your gun and you'll do fine, just don't neglect your gun, either. While Human Sentinels can use Throw almost exclusively, sometimes you'll want Warp to rip through shields or armor. Warp has almost double the cooldown of Throw, and if you have Tech Armor on, nearly triple. Just because you're waiting to set off the Warp + Throw combo doesn't mean you can't soften your target up with a few bullets, too.
If you're playing a Krogan, just grab a shotgun and put the blade attachment on it. You won't be firing it half the time, anyways.
Ignore Banshee's unless you're on Silver or Gold challenge. Just kite them around and focus on killing everything else - Ravagers, especially - before focusing on the Banshee. They are almost completely harmless until they get close to you, so long as that doesn't happen you'll be fine. Ravagers, however, only need one complete volley on any target to incapacitate them, even on Bronze, though they are unusually weak for Armored targets. A mobile-minded Vanguard can Charge right up to one and alternate between Nova and Charging it to kill it without getting hit by it once, with the added bonus of killing all the Swarmers as they spawn. Brutes can almost be bum-rushed the same way, though its a lot riskier; sometimes you'll get snagged by their one-hit-kill technique, despite being light on your feet.
Sometimes your character will get snagged by the one-hit kill skills of the heavy opponents even while putting distance between you and them. This is usually heralded by your character suddenly running backwards diagonally towards the creature. If you're using a headset, just start screaming for your allies to use their Cobra missile launcher on it and hopefully you'll be saved before it completely kills you.
Which, seriously, that's just complete bullshit. There needs to be at least SOME way to fight back against the one-hit kills, not just WHOOPS YOU GOT SNAGGED FROM FIVE YARDS AWAY COME CLIMB INTO MY GIANT CLAW LOLOLOLOL