Before i start, you should understand that these builds are from my point of view, them are not set in stone, other people's views may differ.

Offensive build

Race: Terrix
Stats: at least 30 strength, all the rest in dexterity.
Systems: anything with offense
Weapons: preferable stoned weapons (need blistering steel and poisonous cloud)

At start, this is pretty much all around engineer, not so much of systems/weapons choice. Once you hit 70 tho, you should start farming for money, boss as much as you can for items. ( Items you are after: Paokivs Eye, Dread Plates, Qi Yol, To Dru, Tri Wo, Lun Gi, Xen Ra stones)
Sensor: best sensor for the SA and offense is the Paokivs Eye, good as it is if you can't plus/stone it with more SA/offense/haste, your choice.
Plates: this is tricky, as you need the offense from the Dread plates but also you need hull for survivability ( yes you can repair yourself for 500+ but you need a hull base too ). The higher SA, the better, this will increase damage mitigation and your hull. (1 SA = 3 hull) My suggestion is , go for 2 SA/hull/hull regen plates ( umbrals ) and 2 Dread plates. You can stone later the umbrals with some more SA/offense/HR or haste.
Computer: Look for a SA13+ alien computer, preferable blue, with SA/offense/HR on it, or SA/offense/defense. Same as plates, you can add later SA/off/HR/haste .
Electronic Warfare: same as computer, if no suitable one found, go for Malixs or Shades for the SA/def, stone later for stats of your liking.
Shield: this is a tricky one. If you go solo, you'll want a SA/shields and another stat, most common is defense. This you can stone later with some offense, have it handy when you go alone. For PVP/bossing, you'll want a brick to use as shield. There SA/def shields, stone them as high as you can with SA.
Weapons: for this build, you'll go with 3 plasmas and 1 launcher. Brood launcher is fine , try to get it high plus and stone, leave the effect illusionary foes, good as it is. One of the plasmas can be brekkar plasma, casts poisonous cloud , a crew eater. The other 2 plasmas, can be brood plasmas for the +rep skill, you need them plussed and stoned. One with Qi Yol, you are trying to get the blistering steel effect(pretty much like Burn) and the other one with To Dru , you're looking after Mending Aura (like Tech Advice).

  Your goal stats are around at least 180SA, 200 offense base, with highest HR you can get too.
Mainly your weapons will debuff your enemy (BS, PC, IF) while your offense ensures every shot hits. One of the important stats to stick on your systems is haste, at 100 haste your weps hit 50% quicker (eg. a 30/30 bp will hit like 30/15 ) so it improves your build even more for higher DPS.

Defensive/Regenerating Build

Race: Kalarix or Tekorus
Stats: At least 30 str , all rest dexterity
Systems: Anything with defense
Weapons: Preferable stoned (Armored Aura, Steel Aura, Mending Aura)

One of the easiest builds (and slowest to level) as most systems come with defense  (especially alien ones). So you may gear up and improve as you level. At 70, you'll be able to tank/kill almost anything beside bosses and alike.
Sensor: You are after a high SA/defense/HR sensor, most probably you'll find some SA/def alien one and stone it later with SA/def/HR.
Plates: There are some nice SA35/def alien plates which you can improve even further with SA/def/HR , or you could go for umbrals (HR ones) and stone them with defense.
Computer: Same as sensor, also you could use Brood Brain Mk2 or Torlixe ( high plus desired )
Electronic Warfare: Best i've seen suitable for this is Dreadnaught Augment ( SA20 Hull80 Defense60 ) but you'll do fine with either Shades Wings or Malixs Guide ( for the SA/def ) , if not, any alien EW with enough SA/def.
Shield: Get one of those High SA/def, you're not worried much about shields as you'll be able to stand your ground and repair
Weapons: As you're not looking for high DPS , you want anything with repair skill on them ( brood plasmas or skars ), skars are fine with their own cast ( force blast ), BP's on the other hand , you can improve. Use qi yol stones on them , look for Armored Aura ( kind of Fortify ) and Steel Aura (like Flame Barrier). Also Mending Aura will help with the regens ( as you have your Tech Advice and at later tiers you're getting Hull regen too from the secondary TKR class ).

This build benefit greatly from the SA/def and regens, you're aiming at SA200 and defense300 base ( easy achievable, only from 4 good alien plates you should have SA140 and defense160 ). You'll need alot of patience as you're taking your time to kill any mob, up to 20mins from my experience But you'll do the job. Also at bosses, you'll be the one keeping the merc alive, you won't die as much as others.

General build

Race: Tekorus
Stats: at least 30 str, rest goes to dexterity
Systems: Best SA, off/def
Weapons: Mix of offensive/defensive effects

If you want a well rounded engineer, this should be best to pick, as no matter what you equip ( emphasize SA ) you'll be fine. Race is best Tekorus for the regens, you need all your hull back at all times. Good for solo and bossing alike.
Sensor: could go straight for paokivs eye, if not , get some SA/off/def sensor.
Plates: have a mix of offense/defense plates, as well some hull regen/haste.
Computer: you can find some high sa/hull/HR/offense/defense aliens, any 3-4 stats would do, high SA preferable.
Electronic Warfare: again the SA, you need the damage mitigation to get the most out of reps, so i recommend Shades/Malixs or any SA/def alien with some HR stoned.
Weapons: Best combo would be 1 Skars, 1 brekkar plasma, 1 BP/Skars stoned with Mending or Armored Aura, 1 BP/Skars stoned with Blistering Steel. ( So you will get some good buffs and able to debuff the mobs as well)

Engineers benefit alot from Poisonous Cloud, leaving mobs crewless gives you the upper hand: they won't be able to hit you back (only at tick regen) and they take more damage. So have your brekkar plasma ready. It is advisable to have extra systems available so you can tweak your char for your needs ( more offense/regen when going solo, more defense/regen when going bossing).

Fighting

At lower levels, you'll have a hard time leveling solo so it is advised to level grouped. The Mk1 Droid is not much help, dies quick and the timer is big, leaves you with no repair/heal option. Around lvl40 you begin do to better, can solo from now on all the way to 70. Still slow but nothing is perfect. Watch your crew, you should have medicine up to repair's level, most of the time you should link 1 repair then 1 heal again and again.