I want to use my hunting rifle for my student film in a wooded area, do I need a permit or just notify the authorities?

Although the specific required permits vary from state to state and community to community, it is important to notify both law enforcement and the owner/manager of the area. You will also need to notify/warn anyone who uses or has reason to be in the area on the day or days of shooting so they don't accidentally or deliberately interrupt the production. If you can lock the trigger or otherwise secure the rifle from firing during the production, that would be great; otherwise, check the rifle yourself prior to the start of each shot to ensure that everyone on set will be safe.

You should likely look to the local permit office and get a permit for the filming. That way they will know what you are up to and you can address it when you get the permit.

The answer depends where you are in California. In Los Angeles County, you are required to contact the LA FILM permit office and register all prop weapons with them. This may be the case in other locations, also. Also, depending on the wooded area, you may be required to register with a park or forest system. Call the police in the area where you will be filming and ask. This is very important because there have been incidents of people mistaking a student film for real life crime and calling police. You don't want your actor with the rifle to be gunned down by a swat squad, so make your phone calls and get your paperwork done far in advance.