From the perspective of someone whose work/education is not graphic arts/web design but intersects with it heavily:
An associate's degree with work experience you garnered while earning the degree should land you an entry level job - eventually. Remember, it's an artistic field. Be advised that graphic/web design is a field with a lot of freelancers, and a very high unemployment rate. Bachelor's and Master's in Graphic arts/web design are out there- I'd advise you to consult with people in the industry to see if degrees higher than AA are worth pursuing. In my experience as a photoshopper with a photography BA, I'd say anything higher than an AA is good if you want to be a specialist as opposed to a generalist.
Start at the community college level -- in my experience, schools that specialize in art & design tend to be for-profit institutions that will mire you in debt without offering anything inherently superior to a CC. There's no inherent advantage to going straight to a specialty school versus attending a CC, at least for earning your basic skill set. In fact, amongst people I know, those who studied art & design at the CC level emerged better connected and in less debt than the ones who did the expensive schools. Think beyond the CC if and when you wish to specialize. DO YOUR HOMEWORK and research potential schools throughly - CC and art & design schools both. Never, ever take just their word for their reputation in the industry you wish to go into, or exactly how they'll prepare you for your career. Google them, talk to current and former students, cold call working professionals if you have to - get the scoop before you invest the time, money and lifestyle into a school.
Some specific advice I've gleaned from my experience job hunting in the art, media and design worlds: While in school, obtain at least a working knowlege of the complete Adobe Creative Suite, and as many web design languages you can. Learn it all on Mac *and* PC (cross-platform compatibility is vital in this industry). Employers like seeing a long list of programs you are proficient in. I often see Photoshop, Illustrator, Dream Weaver, and After Effects lumped together as programs employers expect you to know.
I cannot stress this enough: begin looking for internships, assisting jobs or entry-level positions while still in school. The earlier, the better. The closer the work puts you to your desired field, the better. MAKE YOUR SCHOOL'S CAREER SERVICES DEPARTMENT EARN THEIR DAMNED PAYCHECKS, and don't wait until a few months to graduation to engage them. Work with them from the beginning. The more work experience you have, the better off you are. Period. If you come to an employer with both the degree and 2-3 years experience, you're going to have a HUGE edge over the guy who comes to the same employer with a shiny degree and nothing else. Higher chance of being hired, higher starting salary (which can affect your earning power for the rest of your working career). Note on internships: by law, the internship must either pay you or give you school credit. If it offers neither, PASS. If you see a temp-to-hire situation where they want you to work for free/without school credit for months on end while they decide whether to hire you (usually at below market wage), PASS.