Be sure to let Java run, otherwise the game won't load! To change your username, click on at the top left corner and enter a new username. You can play online but can only join 1.5.2 servers.
Once you've joined a server, register your username by entering "/register password password" without quotes to be able to play. Obviously, pick a better password than "password." You will need to do this for every new server you join or every new username you choose.
To play fullscreen, go to Options->Video->Fullscreen: ON. You can also press ESCAPE in-game to get to the main menu. Simply toggle Fullscreen: OFF when you want to stop playing.