Where would be the best area to live and work?

I am 21 years old, Male. I am preparing to relocate in Seattle in the next 2 months. I am seeking work in the Food Services (restaurant kitchen/waiter). I have a car but would highly prefer public transportation and I dont have a dog but planning on getting one once I settle in. I am looking to live close to a cultural part of seattle with a great single atmosphere nightlife including close to grocery stores and a dog groomer. I am interested in Belltown, Capital Hill and Downtown. BUT I am looking to spend $800-$1100.

Any Advice given would be greatly appreciated. Thank You!!