I scp'd my ~/.ssh/known_hosts file over to the server, but it still asks me for my password every time I log in over there (which is quite often actually.) What else do I have to do to avoid having to enter my password every time?
known_hosts is used on the client for protocol 1 connections, to verify that the server is who it claims to be. There's also a known_hosts2 file for protocol 2. On the server you need the files authorized_keys and authorized_keys2 to hold the public keys authorized to connect to that account. Your private key will be in one of the files identity (for RSA) or id_dsa (for DSA) on the client.
Also be careful about permissions. The .ssh directory must not have group or world access. Set it to 0700. The server won't accept PK connections if that directory is too loose.