That depends on what you guys like to do. Here are some options:
If you like to party like a rockstar and want to be with the tourists - South Beach is the place. Mostly tourists, this is the Miami that you see on TV and movies: Art Deco, beach and party. You might see too much of some people at the beach...
It is also the most expensive part of Miami. Most affordable hotels will be kind of sketchy.
If you want to party, but not only - Downtown is a great choice. Bayside MarketPlace (http://www.baysidemarketplace.com/) is the touristic mall with tons of souvenirs, live music at afternoon/night, local and chain restaurants and celebrity
homes sightseeing by boat ride. Mary Brickell Village (http://www.marybrickellvillage.com/) is a half tourist/half local area for bar, restaurants, clubs - mostly local restaurants, with some chains. Downtown can be as expensive as SoBe, but in much
better hotels. Among the most affordable options is the Hampton Inn & Suites Miami Brickell Downtown (http://www.hamptoninnmiamibrickell.com/). It is a beautiful new hotel, #1 on tripadvisor.com, with breakfast included and $18 a day parking
fee.
If you want to be with the locals - Coconut Grove is a nice and quieter area close to downtown. Mostly locals, some tourists. CocoWalk (http://www.cocowalk.net/) has restaurants and bars. I stayed at the Hampton Inn Coconut Grove (http://www.hamtponinncoconutgrove.com/). It is a nice hotel with breakfast and parking included.
Unfortunately, if you want to get out of any area in Miami, you will need a car. Miami is not public transportation friendly. Out of those areas, there is the Miami Seaquarium, Everglades, the Keys, shopping and more.