I played five rounds using St. Andrews as a base back in 2000. We played Old, New, Jubilee, Kingsbarns, and Carnoustie. I'd recommend them all but if I were to drop one out of the rotation it would be Jubilee. We didn't have a tee time on the Old, if you're in the same boat then I'd second the suggestion of scheduling strategically to try and gain as many opportunities as possible via either the ballot or walk up. We tried the ballot twice unsuccessfully but eventually got on by getting to the course very early one morning and we were the second group out (August).
Using St. Andrews as a base is a great idea, the town is a lot of fun to check out and all the history makes it a really fun stay.