For practising specific fights, it's quite helpful to create some new battle groups made up of lots just one kind of ship. (And save them) Here's some tips for beating the ships you have to deal with at the start of the game, in roughly increasing order of difficulty:
Orz vs Slylandro:
This is a stupid fight. Run away and launch 4 marines. The probe will go nuts trying to avoid the marines. While it's doing that, it's not shooting at you, and the marines will get it eventually.
Earthling vs Vux:
Vuxen are fairly easy to kill with the Earthling. All you have to do is keep your distance and fire missiles. Unfortunately, the Vux will sometimes warp in right next to you and slag your ship before you can do anything. There's not much you can do about this

Spathi vs Ur-quan:
The puny Ur-quan Dreadnaught is helpless against the mighty Spathi. Keep out of range, and when it launches fighters, fire BUTTs at them. Or just fly around - the fighters will eventually hit asteroids and die. To kill the main ship, you need to fire BUTTs from just out of its range, while it is chasing you.
Spathi vs Ilwrath:
The Ilwrath can be killed by almost anything with greater range and speed. And that's just about every other ship in the game. You can tell exactly where it is because it's always opposite you. Thrust away from it until you're moving at about the same speed, then turn round and shoot. The Spathi, ZokFotPik and Shofixti can kill it quite easily this way. The ZoqFotPik's scatter gun is hard to miss with (well, hard to miss every single shot with, anyway)
Spathi vs Kohr-ah:
This is much harder than the Spathi vs green Ur-quan fight. Run away from it, dodging the saw-blades and firing BUTTs. Dodging the blades is the tricky part - it's best if the Kohr-ah is chasing you, because then you have much more time to react. You can't outrun the blades, so if one is coming up behind you, turn! It takes a long time to kill the Kohr-ah this way, because half your missiles will hit saw-blades or get F.R.I.E.D. But it eventually goes down.