I guess that anything that has free (rather than tile-based) movement is a good start - in fact you could deal with Breakout's ball using vectors. I would definitely start with 2d rather than 3d.
I'm in a similar position and have written something like Asteroids as a first project. Problems that can be handled with vectors include:
- player vehicle's free 2d movement
- player vehicle's rotation (and thus acceleration and firing) independent from direction of travel
- player's bullets affected by the firing vehicle's movement
- movement of asteroid fragments after a sucessful bullet hit
I have also written AI for the player vehicle; the aiming part of this must take target movement (leading) and own movement into account, which again are good vector problems.