It is a broad and tricky area - we would tend to recommend that you keep your domain registration separate from your hosting so that if anything goes wrong you still know you have control over your domain.
Tips:
As a lot of domains are getting snatcheed up you may struggle to find the one you want so think about:
1. Going for a .org or .net rather than a .com - or look at a regional variation or a .info or .biz one if appropriate.
2. The hyphen is your friend: myspecialdomain.com might be taken but is my-special-domain.com?
3. Try just throwing random words together and seeing if you like anything like:
monkeyladder.com
gravydance.com
4. Slight variations on words or spelling (but try to avoid l33t spellings).