I'm born and raised in Miami and have lived here all my life. Thus, I can't really recommend any place specific for lodging since I've never had to stay in a hotel in Miami. Overall, I have to say that I would not stay anywhere less than 3 stars in Miami since many ppl I know have had very bad/gross experiences with low-rated hotels. Brickell and Miami Beach are the closest areas to the med school, but hotels in Miami Beach can be extremely expensive/overpriced and traffic can be terrible in that area. You definitely need a car to get around miami since public transportation here is not good (the metro-rail system only takes you to a limited number of places, some of which can be sketchy/ghetto). Taking a cab would be the best thing if you don't have a car. I think you have to be at least 25 yrs old to rent a car here but I'm not sure so I would double check. I would definitely recommend renting a car if you can. The places I would visit (in order of distance from the med school) would be:
Brickell (really good restaurants/bars),
South Beach (you can hang out on the beach, weather permitting or you can walk around Lincoln Road; night life here is very good in terms of clubs),
Parrot Jungle Island (a zoo-ish place),
Crandon Park (another beach in Key Biscayne with nice views of the skyline),
Coco Walk in Coconut Grove (restaurants, shops, and bars/clubs at night),
Vizcaya and Gardens (a really nice huge, old mansion that is now a museum with views of the city),
Fairchild Tropical Gardens,
Coral Gables (mostly residential, but very nice, with shops and really good restaurants around Miracle Mile and Ponce de Leon Blvd),
Miami Metro Zoo
For shopping, I would recommend malls like Sunset Place, Dolphin Mall, or Dadeland Mall. You can PM me if you have any questions or want more info! You're not going to have time to see all of the above unless you're staying for about a week so I would stick to what activities you would like to do most and refer to the above. Also, I don't know how long you're staying but avoid driving during rush hour on weekdays if you can. Traffic all over Miami gets impossible on weekdays from about 8am to 10am and again from about 4pm to 7pm. Hope this helps!