I've worked at gaming companies, and although I never worked on the games themselves (more the backend, console, etc), I would suggest just writing some games yourself!

There's a lot of aspects to game development, including gameplay, art, storyline, and many of those aren't technical. If you do get into a technical problem that can't be easily solved, cheat! Many games have an addictive mechanic that comes from a hard technical problem you want the player to solve.

There are some frameworks for writing games, and you can get familiar with whatever you think will solve the problem.

Also, just apply to a game company! They need web people too, and it's easier to use your skills to get your foot in the door, then you can learn from the inside.