Been a while since I played, but…
Shields to 2 is a priority as it's enough to block most earlygame laser and beam fire. After that, engines. Shields to 3 some time around the 4th sector. I rarely bother with 4 layers of shields. 3 is enough to block any single weapon, and at a certain point the ~60% dodge rate from high level engines becomes far more valuable than one more absorbed hit.
If at all possible, you should be aiming to defeat enemy ships by killing the crew rather than destroying the ship. Plenty of ways to do it, but personally I like a combination of hacking, fire weapons (preferrably Fire Beam) and mind control. Hacking is versatile to the point of being one of the best subsystems in the game… I'll typically use it to either drain the oxygen, keep doors locked while fires spread, or if the enemy is well armed, to keep their weapons under control. More rare for me, but it's also great for taking down enemy shields if you lack another reliable way of doing it. Fire weapons are great for both dealing with crew and keeping annoying/dangerous enemy systems offline. Mind control I mainly take just to keep my own guys from being mind controlled against the flagship, but it can still contribute leading up to the endgame. Enemy ship has a high evasion rate? Mind control the pilot before firing. Enemies trying to put out a fire? Mind control one of them to put an end to that. Repairing a system I want to stay offline? Just low health and I want them to die? You get the idea.