Building an Engineer for Multiplayer
Paralytic's answer is pretty accurate. As an engineer, you should not be relying on weapons but on using your tech abilities for maximum damage. Personally my favorite is the Salarian Engineer, but the following tactics will also work great for the human engineer:
Your bread and butter damage skill will be incinerate. You should be focusing on single target damage rather than AOE with this skill since it is especially effective against armored targets, and the tough enemies all have high armor. Also, having high damage will let you one shot the usual minions. A brute should die within 3-4 incinerates if you shoot him with your pistol in between recharges.
Energy Drain is an excellent survival tool, allowing you to keep your shields up when under heavy pressure. Against mostly geth maps, this will be your primary damage dealer, you can easily mop up entire maps spamming this and causing a tech explosion with a incinerate to follow up. I went with max damage, but if you want survivability the extra armor/shields can also come in handy.
You have the perfect tech burst combination in edrain and incinerate. Even against fleshy enemies, edrain can stun them for several seconds, and it can set up a tech burst if you follow it up with incinerate for a lot of damage. You don't even have to kill a target for this tech explosion to work, you just have to edrain to shock them then follow it up with a incinerate.
I found that 1 point into decoy is plenty. With extremely fast cooldown, you can keep this up and spam powers none stop with no problem. Decoy is an excellent skill that can draw a lot of enemy fire. Even better, its a portable barrier. If you are facing an geth prime with no cover nearby, putting a decoy between you and the geth prime will mean all his shots won't touch you at all. Decoy is also great for those situations where you have to camp out in a zone for a long time. You can easily hold an entire corridor with decoy and some tech explosions, especially against the geth.
As a skill spammer, you shouldn't be carrying around a lot of weapons. A light machine gun and/or pistol should be all you want. Try to keep your power recharge to as close as 200% as possible. I go with a good pistol with the scope and damage mod for the guardians (they are the only class that can withstand your powers due to their annoying shield). Try to head shot them with your pistol.
For human engineers, overload replaces edrain and the drone replaces the decoy, note that the drone does not draw fire, so its not as good for defense, but it can turn guardians around, which is handy.
Finally, for Quarian Engineers, don't underestimate the cryo blast + incinerate explosion combo, it is great crowd control.
My score with a salarian engineer is always the highest in the games due to him being able to handle every tough enemy with ease.
So, I roll with a human engineer most of the time, and very rarely finish as anything but the highest score, especially in silver and gold matches. I upgraded incinerate, overload, and combat drone to the max, while putting 5 into N7 operative (or whatever it's called), and 3 into fitness. Admittedly, this makes this build "softer" than others, but the damage output you get from abilities is absurd. Some general things about the engineer: as many people have mentioned above, the name of the game is your abilities, and how fast you can use them. Basically, if you don't have the Carnifex pistol yet, get it. Now. No excuses. It's a headshot machine, doesn't really affect recharge time (you get %160+ recharge speed), and is generally one of the best all-around weapons in the game. You can roll with the phalanx or the Paladin if need be, but I prefer the Carnifex as a good medium.
Now, I'm gonna assume whoever is reading this played ME1 and ME2, where the Overload ability kind of blew. Like a lot. However, if you take the chain lighting-type upgrades plus the upgrade that incapacitates organic enemies for a short while, the engineer becomes one of the best crowd-control classes in the game. It stuns everyone (the first enemy hit actually falls to the ground or is staggered and takes a knee if they had shields) it hits, including those enemies the overload "jumps" too, and allows you to set up easy headshots, while none of them return fire, so important on higher difficulties, where apparently your shields are based off a butter-like substrate.
Combat drone is great for distractions (why hello, banshee, play with my friend for a while), and should be upgraded to cause massive close-quarters damage (so go for the explosion upon death and the close-range stun move), plus the rocket. I mean, it's a friggin rocket,it does 400-some damage, has a large radius, etc. It'll die quickly on higher difficulties, regardless of how much health/shields it has, so make it do what damage it can while it's there. And I know people hate on the drone, but I can't stress how many times it's saved my ass on silver/gold matches. Deals a solid chunk of damage while distracting people, thereby setting up further punishment. And then you can keep. On. Spawning it...cause it's recharge time is faster than overload/incinerate.
Incinerate is great against big, boss-type enemies with armor. Don't take the upgrade to increase radius over damage (remember, overload should be your crowd-control move), and make sure to increase damage against armor by 50% instead of the increased damage to chilled/frozen targets, which relies upon another team mate having cryo blast or someone having cyro ammo. It's just not worth it. Also, take the level 5 upgrade for increased damage over time, this'll make you absolutely truck those big obnoxious enemies, and is pretty much what I only use it for, due to a longer recharge time as opposed to overload.
I cannot stress how gnarly this build is, only an asari adept is as good as the engineer in my book (biotic explosions...so unfair). Furthermore, the salarian engineer is pretty boss as well. Energy drain = infinite shields, my friends. However, that being said, the combat drone serves basically the same purpose as the decoy ability...except that is deals out more damage, you can spawn it at any enemy's location on the map (provided that you're locked on to them), and you don't have to deal with return fire...so clutch.