Should I prefer hash algorithms with longer outputs for storing passwords?

Question

I'm building a site where security is somewhat important (then again, when is it not important?) and I was looking for the best way to store my passwords. I know that MD5 has issues with collisions as well as SHA-1, so I was looking into storing my passwords via either SHA-256 or SHA-512.

Is it wiser to store a longer hash variant as opposed to a smaller one? (ie 512 vs 256) Does it take significantly more time to crack a SHA-512 encoded password versus a SHA-256 encoded password?

Also, I've read about using "salts" for the passwords. What is this and how does it work? Do I simply store the salt value in another database field? How do I use that as a part of the hash value calculation?