Best game evah. But I first fell in love with the 3DO version (which was actually the superior version), so when first playing Ur-Quan Masters a couple years ago when it came out, I flat out sssssucked when trying to control my ship with the arrow keys. I never checked if they added joystick support.
Anyway, yeah, what they said. Mine the nearby planets and systems first, maybe for a half hour/hour or so, sell those minerals and build up your ship. There's really a lot of room to upgrade it from the useless boat you start with to a pretty sleek, powerful ship once you max it out.
I thought the 3DO version never made it to retail! The one Patel mentions is, I believe, the executable of what would have become the retail version for the 3DO.
Anyway, great game. Amazing game. I played this with some buddies back in the day and we'd share strategies on how to pass certain parts of the game. The "Star Control-ese" we'd speak was frightening, borderlining on trekkie nerd fanaticism.
Apart from the Wiki's suggestions, I'd focus almost completely on building your mothership's speed, then turning rate (probably 2:1 or 3:1). You will meet with some badasses and you will not be able to defeat them, so the best thing is to be fast enough to outrun them and avoid conflict altogether until you are able to build up on weapons.
Avoid the Ilwrathi, Kohr-Ah and Ur-Quan like the plague at first. Only when your mothership is powerful enough do you want to engage these bastards.
Basically mine and upgrade first, then travel to other regions to follow the plot, and finally focus on battling the enemies (the ones that will never under any circumstances ally with you).
Don't even bother with gas giants (they are the large planets) - they never have anything to mine. There is one gas giant with an alien race - I can't remember the system but you can tell because normally there will be alien ships present in the solar system view.