Heather M’s Answer
You can technically take either path. I earned a two-year graphic design degree at my local community college and it was much more affordable than if I had gone to a comparable university or private art school. Do a lot of research into the programs taught and course content before you decide. In the end, it's not about whether the school is an art school or a big name school or any of that: it's about the skills they're going to teach you. Look at real job listings you're interested in and be sure the degree you're considering is actually going to teach you the specific programs and techniques required in the real world.
A good graphic design program will have a portfolio course of some type, regardless of whether it's an art school or not. That being said, you will always end up altering your portfolio after you graduate. For each job you apply for, you'll want to tailor your portfolio accordingly. For example, for a logo design gig, you'd want to showcase all of your logo and Illustrator work rather than cloud your portfolio with web design and print layout pieces.