Santa Cruz doesn't really have a bad time to apply for apartments. There are also enough rental apartments to go around at any given time. A good suggestion is to just give yourself between six and eight weeks no matter what time of the year you decide to apply at. This should give you ample time to find an apartment.
Santa Cruz landlords are used to dealing with high-income earners that work for the numerous tech companies in the area. Asking for a credit check and proof of employment or income is something that you may have to contend with. If you're unable to meet the rental requirements on your own, ask if you can apply with a consignor on the lease, or if you can apply collectively with roommates for an apartment.
The Santa Cruz Metro public transportation service is actually one of the better in California. It provides excellent connections between the University and downtown. Cycling is also very popular, however the mountains in the area can pose a pretty serious cycling challenge to those who aren't physically fit.
The Beach Boardwalk is a popular Santa Cruz hangout. You can get out on the Boardwalk for free, and the rides are a few dollars apiece. Watersports like surfing, sailing and kayaking are very popular, and the local mountains provide a very popular playground for mountain bikers. Pacific Avenue is the place to go for nightlife, where you can find both upscale bars and lowly pubs.