I missed 2 things from those screen shots. First I missed that the game was indeed paused
Second, I missed that YOU ONLY HAD 2 ENGINES.
So here's something I've posted a lot recently but hey, if it works it works:
Before the end of sector 1, 2 shields (4 bars)
Before the end of sector 2, 4 engines.
Before the end of sector 3, 5 engines.
Before the end of sector 4, 3 shields (6 bars)
Note: This is for NORMAL mode. For HARD mode, I try to shift everything up one sector though it's pretty much impossible.
Also, you need to have a way to get through:
Before the end of Sector 2: 2 shields (Most ships can do this)
Before the end of Sector 4: 3 shields (This usually means you need to buy weapons or a teleporter in sector 3)
Before the end of Sector 7: 4 shields (or the Boss is gonna suck)
Note: to get through 2 shields, you need to do 3 laser shots. Or have 2 ions and a laser/beam/whatever. Or have a teleporter. Or a bomb. Doesn't matter what.
After I have the above, I start thinking about upgrading other things, like doors and O2 and medbay and stuff.
Most of the time I don't reach these milestones, and I end up not getting my 4th engine until midway through sector 3 or something. No biggie, as long as you keep trying.
Never, ever, ever, EVER buy missiles. Unless you simply must because you've not found a way to get rid of your missile weapon. They are the single biggest resource drain in the game if you buy a lot. The next big resource drain is repairs. Any time you show up at a store needing a ton of repairs, that's a weapon you can't buy, or drone control or a cloak or long range scanners or who knows what else. I never repair over 20 hull points (2/3 full) and sometimes I forego repairs to buy or upgrade something.
Ions are great in groups, but not so great by themselves. 2 Ions will eventually take down ANY shield so long as your opponent doesn't dodge too many of them. An Ion and an Ion 2 will reliably take down ANY shield eventually. Once those are down, you can use your other weapons to actually damage the ship.
If I may humbly suggest, I've beaten the game on camera with every ship in the game (and a few mods) on Normal and while I get into (and out of) some sticky situations, for the most part I walk over the bad guys because I stick to what I've said above. Granted, most of my videos are in the previous version but most of the strategies still work. Check out my thread here