What file sharing protocol should I use with an older Mac? I understand that Apple has switched to SMB2 as their default file sharing protocol these days, but I have an old Mac at home that I want to share files with, and it can only run OS X 10.6.

OS X 10.6 gives me the option of using AFP, FTP, or SMB (version 1, I believe).  Is there a reason to choose one of these over the others?
AFP is Apple File Protocol and has been the default for a long time, and will be the most compatible with your 10.6 computer, and probably the easiest to set up. If you are sharing files on your own home network, choose that. It is still compatible with 10.9, it's just not the default network protocol in 10.9. SMB, the common Windows protocol, is available, but does not support some of the "nice" features of Mac OSX; sometimes it won't display custom icons, and might not fully support some file resource forks. FTP is more common for internet based networking, and I see no reason to use that in a home-only setting.