don't go to far north, it's a bit of a ghetto.
I stayed on miami beach which is like 40th street or 50th street, it was nice, I stayed in a hotel with a few people. The beach there was completely empty, very beautiful. any farther north and it got a very dodgy very quick.
it was a $2 bus ride down to south beach where all the action is. South beach is like 5th street to 12th I think there are some hostels there. look on hostel world or on this site for budget accommodation. The beach it self in south beach is a bit dumpy. It seems fake like they truck sand in and pour it over concrete. but I'll be damned if south beach is not the most fun place on earth filled with the most beautiful people. Heaps of great food down there too!
South Beach and Miami beach are both a bit of a drive from from the airport. the airport is more on the main land part of florida. look online for a shuttle bus there must be one.
I rented a car and drove to the everglades, and did an airboat ride which to this day is one of the coolest things I've ever done. It's just awesome, it's just wetlands for as far as you can see! heaps of aligators too! You would absolutely need an entire day and a car for that.
I suggest renting a car for a day or two because miami really is a beautiful city with a ton of culture. It's really cool just cruising around the bridges and highways blasting spanish radio stations!