Baton Rouge has a good selection of available apartments at any given time. However, the city is large enough that location matters. To have your pick of where you want to live, starting to look in the early summer is going to yield you the best options.
When you're applying for an apartment in Baton Rouge, tell landlords up front if you're a student. Some landlords will rent to students and some prefer not to. However, if you hide that information and then tell them you're a student when asked to provide proof of employment later in the application process, you can be sure that you've already sunk your chances.
Public transportation in Baton Rouge is operated by CATS, the Capital Area Transit System. CATS has synced its tracking system with the Route Shout app so that smartphone users can see exactly when and where busses will arrive, making it a very user friendly system. The app is free, and the addition of Wi-Fi on certain bus routes makes this even more accessible.
Baton Rouge contains the trappings of a Capitol city. The State Capitol and the Louisiana State Museum are well worth visiting. If you're not from Louisiana, be sure to get outside of the city and experience the Bayou. Tours are available that will take you out in the swamp where you can see alligators and other Louisiana wildlife in a natural habitat. To complete the Louisiana experience, you need to get some authentic food. Try the Zeeland Street Market for affordable, authentic southern cooking.