Hi everyone,
I played SC2 back in the 90s, and I've been looking for it for the past 10 years and finally found it on GOG. So, that's good.

However, I'm terrible at the combat. Seriously. I can't reliably kill anything without using the Precursor ship. Every now and then I can take down an Ur-Quan, VUX, or Mycon with an Earthling Cruiser. I don't think I've managed to win with anything else, and so far I've tried using Spathi, Arilou, Zoq-Fot-Pik, Orz, Yehat, Pkunk, and Syreen ships. (The Spathi in particular spelled disaster because I never could figure out which end was the front of it.) I also keep getting blindsided by the planet -- I'm on the edge of the screen, and next thing I know I'm crashing into the planet that I couldn't even see until the collision. Is there a way to avoid that?

This is very unusual. I'm very good at video games, including dogfighting games and others of the drive/fly-around-and-shoot-at-eachother variety. I understand the principles of dogfighting: leading your shots, closing to optimal range while keeping out of your enemy's optimal range, approaching from a direction that the other guy can't shoot in, etc.

But I'm still no good at shooting ships in this game. I get blasted. So, clearly, there's something I'm missing here. I'm approaching it the wrong way. Can anyone help me out here?

I've more or less the same problem - gave up on SC games because of it. I can't offer much advice, but, I wasn't half bad with the Spathi ships. Tapping the forward key a few times (just enough to show the trail) will help you figure out where's the front, and with its homing missiles you should have a fair chance against almost anything.
The basic tactic is to let the enemy pursue you, and plant a rocket in their face when they gets close enough. The right distance to do so depends on what you're up against, if you shoot too early the other ship will just turn and outrun the rocket, and if you wait too much you'll be exposed to enemy fire.
In general, I found that a good time to shoot is right after the camera zoom activates. Slowing down the game in DosBox also helps.

Combat can definitely be a little unforgiving. Personally I've found that the best thing you can possibly do is simply to pick the right ship for the given situation, as going up against a Slylandro probe with, for instance, a Zoq-Fot-Pik is a waste of time and resources, no matter how good you are. Sadly, the only way of knowing which ships are effective against which is by experimentation (which therefore involves, most likely, lots of failing / dying).

In addition, never be too proud to run away from a fight. Some fights are either best avoided altogether, or avoided at least until you find the right alien race with which to combat them effectively. You'd surprised to find just how few fights you actually have to get into during the course of the campaign. Although obviously you want to have a few scraps here and there, since that's part of the fun.

That said, I've noticed as you have that the earthling cruiser is surprisingly versatile ship. So don't be afraid to use it.

Then again, I'm also of the opinion that some ships are hopeless regardless of the situation. Period. FIghting with them is usually just an exercise in frustration. The Shofixi is one such case, as are the Syreen, the Illwrath, and the Arilou (to name some). Just my personal opinion though. Arilou *can* do good damage if you're fortunate, but for me those situations are too few and far between to make it worhtwhile, and they are never certain since you never know where you're teleporting to in the first place, and second, the high speed and insanely sensitive turning rate mean that you sometimes actually steer into projectiles while trying to avoid them.

Best way to fight - fully upgraded Precursor ship, with 150 (or even 200) crew members.

By fully upgraded, I mean ALL thrusters, ALL Turning jets, Front and Spread Hell Bore cannons, 3 0r 4 Shiva dynamos, Automatic tracking modules (2 or 3), and Point Defense for fighting the Ur-Quan ships with fighters

I started to win more often when I switched to gamepad.