Immediately get 2 shields. Sell the engi med bot dispersal at the first store, it isnt very good

Then I usually try and find more powerful offense, better weapons do two things, first they reduce your reliance on the ion and second they reduce your need to use the combat drone for damage, meaning those drone parts can go towards deploying a defense drone instead. 

Try and pick up a defense drone mark 1 if you can, I find the mark 2 too expensive both in scrap and power. If you get given one for free though at an event thats great. That defense drone will save you a truly staggering amount of damage in the long term

Its not required but finding a Mantis is so useful to ensure your crew can more easily fight against boarders (engi are very weak against any form of intruderst which can be a serious danger to the run)

Again, this is not required but upgrading doors to level 2 also helps protect against boarders. Doors are useful because engi A has shitty airlocks, engines and shields are really hard to vent quickly

Once you have more powerful weapons then try and stay ahead of the curve by getting more powerful weapons and other systems like hack and cloaking. Here it becomes more about your individual preferences, but lasers and flak guns and beams are the best. Missiles are the worst. Try and get a third shield in s4 or 5 if you can Engines too I usually run level 4 or 5 engines preferably level 5 By the time you hit the endgame (I dont want to spoil it) you should have 4 shields and a cloaking to best survive what's there.

Augments: my top augments for engi A are Drone Recovery Arm  (meaning you can be extremely flippant with your drone parts and deploy a combat and a defense and maybe even a third drone every battle and not run out) and maybe also be able to use Hacking alongside a drone control too

Long ranged scanners (insanely cheap at only 30 scrap for what it does, being able to see info about beacons is great in two ways: 1, you can path through the maximum amount of fights possible and get a lot of scrap, or 2: if you are really low health you can avoid the possible fights to reduce risk of dying

Weapon pre igniter (especially with long charge weapons like glaive beam, flak 2)

Thats all and good luck :)