As one of the two Threewave Capture the Flag mod programmers for Quake 3, I can testify that he is working on (and sending) the mod as an incredibly useful introduction to games.
Working on fashion places you next to the production code with all its conventions and warts. It also introduces you to an existing code base, which is at least several hundred thousand lines of code. This teaches that you can search and read code and are easy to work in a team.
You can also work with assets of quality art and deliver a product that attracts attention for you and your small group.
Working with XNA or downloading libraries and chatting on forums is fine, but none of them give you a starting point for a specific product that is already commercially successful.
Half-Life 2 seems like a great starting point these days. A lot of C ++ code, community support and the history of Valve and other studios hiring the best talents from their own community.
Good luck
source share