If you want to race for the University, then you're in luck. The CUCC is active in pretty much all disciplines - road racing, time-trialling, hill climbs, mountain bike racing.
Time-trialling is a full blue sport and mountain biking a half blue sport. If you want to check the qualifying criteria you can do on this section of the website. If you're an established racer and feel like seeing whether you can make some club history, then take a look through the club records below.