scombinator is certainly right, SQLite is really nice.
Databases are made (mostly) for variable sized data that needs to be searched so put players and stats (and why not savegame data?) in there but things like maps would probably fit better in a 'binary' format like a file.
I have used MySQL and SQLite and a database is really handy to have, you can just make a new table and stuff mails, logdata or whatever in there.
BTW. I'd recommend MySQL is for server side (multiplayer / connected games) or SQLite for embedding in your game (singleplayer).