Because the school is very much the focus of the entire community, rental availability strongly follows the flow of students in and out of town. Many students will give their notice in February or March, which is a good time for new students moving to town to start looking.
Getting an apartment near Stanford is nearly as competitive as getting into the school. Stanford University makes a few recommendations for finding off-campus housing, one of which is using a rental service. The environment is competitive enough that if you don't use a service, you may find yourself at a disadvantage when compared to renters who are.
The Stanford campus is a public transportation hub that has train and bus services that connect to other towns such as Palo Alto in the immediate area. If you're staying within Stanford proper, you will likely find that a bicycle or your own two feet are more than adequate modes of transportation. The weather makes them both viable year-round options as well.
Stanford itself provides some of the best entertainment in the area. The University is home to a number of arts events including live music, theatre, dance, and numerous arts exhibits. Stanford Cardinal football program competes in Division I NCAA football, and the turnout at Cardinal games is fantastic, with the stadium having room for 50,000 spectators.