I currently live in Royal Oak on 14 mile and Coolidge. I don't live downtown though, but I'm pretty much within 30 minutes of everywhere in Detroit. Grocery store, restaraunts, target, kohls, home depot, etc are all about 1 mile from my place. Downtown Detroit is somewhere b/w 15-30 minutes depending on the traffic. It's nice b/c you can decide to take I-75 or Woodward all the way downtown. Woodward being a pretty constant 20-25 minutes downtown. I-75 is hit or miss depending on weather, accidents, events, etc.
I'm in Public Accounting so I'm driving all over Detroit area (within 1.5 hours of my apartment) and I'll tell you it's the best location traveling wise. I'm always going the opposite direction as the busy traffic. If you live on the west side (Livonia, Novi, Canton) you will hit monster traffic going into the city in the morning and coming home at night. Your commute will be a good 45mins to an hour. More if the weather is bad. East side of I-75 (warren, etc) are ok. They are not as nice as RO/West Bloomfield/Birmingham, but are cheaper to live in.
For the gym, I workout at Lifetime Fitness. I got a good deal for 40 bucks a month and it's open 24 hours. Its about 5 miles north of RO. There are other smaller gyms around too, but I like the idea of the pool, 24 hours, and free aerobic classes if I want to change things up. Plus they do free fitness assessments (bf%, strength, flexibility, VO2 max, etc.) and I can use my membership at any Gold or lower level Lifetime fitness.