Annalee - Advanced reservation for St Andrews Old Course normally occurs in late August through mid September. Other than the Old Course there are 6 wonderful course associated with the St Andrews Links to play. Check out www.standrews.com.
Also as others have mentioned there is the ballot(I've had success with the ballot) and arriving in the wee hours and maybe the starter will be able to get you on(I've had success with this also).
On other golf thing to do in St Andrews is the Himalayan Putting Course. Its only £3 and is loads of fun. Check out www.standrewsputtingclub.com
One other thing I forgot is the minimum handicap requirement to play the Old Course 24 for men and 36 for women. I have never had my handicap card checked, but I did carry it with me.