Open is the normal game. "Conquest" has a few bonuses for easier starts (treasure chests you can disband for extra shields, maybe extra units too). Predator is supposed to be the hardest.
Conquest players have a reduced score (15% I think) but Preds don't get a bonus for a tougher game.
Everybody needs the install - files for extra civs and resouces - but the save start positions are different for each level.
A good place to start is to make sure you Civ or PTW is patched to latest versions (1.29 for Civ3, 1.21 I think for PTW).
Don't be afraid to download the current game - even if you don't finish, you'll have less to download for the next one. Plus you'll be more confident for when you start on the first one you intend to submit.
Oh yes, us 1.29 players have a lot of extras added for PTW compatibility (Stock Exchanges, ports of Entry, Intenet Wonder etc).
Like the guy said, download and Enjoy! Expect to learn lots - I have in just 4 games! The install is automatic too now - but have a pen handy for noting the addresses for the save.