[Suicide Junkie]

Once you have an account, log in.

In graphics mode, you can mouse-over all the buttons to see what they do (text appears on the right)

Click the "attack" icon to go to the games screen. By default you see your current games, colour coded by status.
Click the fleet icons to view "open games" or "all games", and then choose one from the left.

When you see a game you like, select it by clicking, and then click the resupply-at-nearest lightning bolt to join.

During the game, you use the cargo-load and cargo-drop to upload your turns (and empire file) or download the next turn.