University, hands down. Most film schools are heavily project-based, which allows you to gain experience doing real-world projects while under guidance of experienced professionals. There is a lot more to good film and media production than most people think at first, and I know that I have benefitted immensely from my experiences in a university program. It might seem a little irritating to put your professional dreams on hold for a few years, but you will have much more success when you actually begin working.
Something that is less available in my very small program but is certainly present in larger schools is an extensive network in the industry, allowing you to easily slip into internships and jobs, or alternatively get quick access to new clients and technologies during and after your schooling.