Some games to try to program
Chomp: http://lpcs.math.msu.su/~pentus/abacus.htm
Northcott's Game: http://www.cut-the-knot.org/recurrence/Northcott.shtml
http://www.cut-the-knot.org/Curriculum/Games/SilverDollar.shtml
http://www.cut-the-knot.org/blue/tt.shtml
http://www.cut-the-knot.org/SimpleGames/FrogsAndToads.shtml

Fang - an engine that supports two players
http://www.fangengine.org/images/docs/api/
http://www.fangengine.org/index.php/Download
http://www.fangengine.org/index.php/Tutorials
http://www.fangengine.org/index.php/Tutorial:Wackadot

How to program game java applets
http://javaboutique.internet.com/tutorials/Java_Game_Programming/
http://java-programming.suite101.com/article.cfm/an_introduction_to_games_programming_with_java