Unfortunatly, there's no universal guide or flowcharts to troubleshooting game boards that I am aware of.
How familiar are you with electronics already ?? If you aren't already into electronics, I'd start by hitting your local bookstore or amazon and find some books on basic electronics. Then to more in depth subjects such as how logic works, cpu's function, bussing and addressing systems operate and so forth. You'll need to have a grasp on logic probes or better yet, an o-scope.
There's many repair logs and guides available on the net for specific games and problems associated with them, those are often very helpfull and you will learn alot.
Have fun!
D