Morgoth was recorded as the most powerful of the Valar, at least at the beginning of the Silmarillion.
Power begs for corruption. It went to Morgoth's head and created within him the desire to be even more powerful. This need for strength is sated somewhat by dominating others, and that is exactly what Morgoth did, by corrupting the elves and creating his own little minions to do his bidding.
After that, it's only natural that he would want to dominate everyone else as well.
As for Sauron, well, his name before he joined Morgoth was Thu, and if I had that name, I'd be evil too.
(*ba-doom-ching!*)
Anyway...
He most likely wasn't a very important little Maia, and wanted some recognition. So, he basically followed Morgoth's example.