Yes, you can mix brands--and should, since the best speaker companies tend not to make electronics.
The most important thing to remember in assembling a budget system is that speaker quality is paramount. So put most of your budget there.
I would not buy a combined amp/disk player. Ideal for a budget system would be a new stereo receiver (Onkyos are great bargains) or used integrated amp, and a plain ol' DVD player. That should be doable for no more than $300, leaving $400+ for speakers.
For newbies, I don't recommend buying used, since you don't know much about the market yet. For an amp it's ok (although I personally think the budget high-end brands are overrated). But never buy speakers without listening to them first, which rules out most used options.
Speaker brands to look for include PSB, Paradigm, and Infinity. But there are many good options out there. You should listen to what's available in your area, and buy what sounds best to you.
If you can't explain how it works, you can't say it doesn't.—The High-End Creed