A lot of this is dependent on class. As a Vanguard for example, you'll probably never want to use an Assault Rifle - they're just too heavy, and as a Soldier, you'll usually give SMG's a pass - everything they can do an AR does better. Meanwhile, an Infiltrator gets lots of bonuses when using their trusty Sniper Rifle, evening if irresponsibly shooting it from the hip.
The second consideration is range: at very long ranges, a Sniper Rifle offers top notch accuracy. AR's and Pistols Can be pretty good. SMG's can be flaky though, and shotguns are nigh useless. By contrast, Shotgubs ate devastating at short range, but the loss of peripheral vision while looking down that sniper scope can get you killed.
The next consideration is the defenses of your opponents. Slow firing, high caliber weapons like Sniper Rifles and Pistols chew through Armor, but are weak against Biotic Barriers and Shields. Rapid fire weapons with a lot of projectiles, by contrast, like SMG's and Shotguns chew up Shields and Barriers quickly, but have trouble with armor. Assault Rufles are a bit of an odd duck as they're 'pretty good' at handling everything.
Additionally there are mods availiable for several weapon types that alter their capabilities. You can add Armor-penetration to an SMG or a shotgun, or a melee upgrade to a pistol or shotgun. Maybe a scope on your SMG to improve its mid-long range performance?
Finally, you may want to pair some weapons up with certain specialty ammo. For instance, SMG's are devestating in combination with Cryo or Incendiary Ammo as their high rate of fire gives them more opportunities to proc the special effects provided by these ammo types than, say, a slow firing shotgun.
Lets take a look at each bonus power and break it down individually:
Ashley (assuming she's alive):
- Inferno Grenade: Not a bad choice as the engineer class doesn't have any grenade powers. Really useful to flush enemies out from behind cover.
- Marksman: Boosts accuracy and fire rate of your weapons, but an engineer is more focused towards powers so not that useful.
James:
- Fortification: Provides defense and melee on purge, but slows power recharge by 60% (can be reduced to 30%). Although survivability is nice, an engineer relies heavily on power spamming so the slowdown on recharge is a big killer.
- Carnage: Very high single target damage, can even stagger Atlas mechs. Pretty useful skill in a pinch, does high armor damage, but incinerate also does high armor damage.
Kaiden (only if he survives virmire):
- Barrier: provides defense at the expense of power cooldown just like fortification, not really a good choice imo.
- Reave: Insanely useful as the only other way to recover health is to use a med-gel.
Garrus:
- Armor Piercing Ammo: Since the engineer doesn't have an ammo ability, and the majority of big and tough enemies in the game high high armor, this is an excellent choice to take.
- Proximity mines: Very handy if you can lay down a lot prior to an engagement. Enemies tend to be more aggressive and mobile on insanity so this isn't a bad choice.
Liara:
- Stasis: Holds a single (non armored) target in place for a long time. Okay ability, not too useful imo.
- Warp ammo: One of the best ammo powers imo, grants high damage against everything.
EDI:
- Defense Matrix: Same with fortification, reduces power recharge as a trade off for survivability.
- Decoy: I never played around with this much, so I can't say either way if its effective or not.
Tali:
- Defense Drone: Having an additional target on the battlefield is always handy. Fully upgraded drones also have a handy AOE effect in chain lightning. I like this skill a lot.
- Energy Drain: Does massive damage to shields and barriers while recharging your own. This is a great skill to have since it provides survivability without suffering power cooldown penalties. This is a good alternative to overcharge.
Javik: I don't have him so I'm leaving him out.
In conclusion here are the good ones to consider:
Reave/Energy Drain for survivability
Defense Drone (and possibility decoy) for an additional target
Warp/Piercing ammo for an ammo power
Inferno Grenade/Proximity mine for something different
Best Answer
Definitely go with the shot gun, I've tried the tempest and it just takes waaaaay too many hits to kill anything, not worth the trouble. Granted I've had a lot of fun with it with the Turian soldier's Marksman ability to put out an absurd number of bullets, but it just doesn't kill fast enough.
As for the shot gun of choice, I really like to geth plasma shotgun. It does a lot of damage, can be charged (you can still use abilities while its charged!) and it has a homing ability so if your crosshairs are on a target, even if its really far away, your shot will track and land on it. As for mods, I go with the increased damage and accuracy mods.
The only drawback is its low ammo count, so plan accordingly.