Go to Community College FirstDepending on the culinary school you're planning on, you might be able to save A LOT of money by taking the pre-requisites at the community college.
If you can transfer in credits for Math, Science, English, or other pre-requisites that cost much less at the CC, then you won't have to pay a much higher per-credit price at the expensive culinary school.
This will also give you time to start working somewhere and decide if this career is really for you, before you start spending tens of thousands of dollars on the "name" school.
I've said it MANY times in this forum, I've never asked someone for their diploma during an interview. I ask them to make a sauce, and ask questions about their work ethic. I'd much rather train someone on cooking than try to teach them the basic things that their mothers should have taught them years before. Showing up for work, being a student of your career, treating others well are things I can't teach an adult.