I've lived in Miami for the vast majority of my life, though I now spend most of my time out of the country, so I'm qualified to answer your questions.
As others have mentioned, Coral Gables and Pinecrest are both good choices in Miami. Pembroke Pines is also a decent choice if you are looking to live in the far north Miami / far south Ft. Lauderdale are. Personally, I find North Miami and Ft. Lauderdale to be boring, but others will disagree.
Miami Beach has the highest number of clubs/hang-out spots of any area in Miami, but central Miami (Kendall) also has a few places to go, as does Coconut Grove. Both are also decent places to live if you find the right neighborhoods.
An oft neglected area of Miami, and thankfully so, is the deep south part of Miami. Just south of Miami is Homestead and Florida City, and there is also the far east side of southern Miami - neighborhoods along Old Cutler Road. This area of Miami is not like the rest. English is still the common language, though Spanish is not rare. Crime is low, population thin and traffic light. But you are still in short driving distance to Miami - say 30-40 minutes to downtown Miami on an uncongested day. Another novelty of this area is that you can drive to the upper Keys in approximately the same amount of time. Housing costs are relatively low in this area, but rising very, very quickly. And like any other area, the specific neighborhood is important. West homestead is quite bad, while east Homestead is quite nice. Also the very far west is the Redlands, which are very safe and you get a lot of property for your dollar.
It's all in what you want. If you want to be in walking distance to clubs, live on Miami Beach. If you want peace and quiet, but to be a relatively short drive to Miami, pick a place down south. If you want a neighborhood that's a bit noisier, and where you're a really short drive to the action, choose east central Miami.
Miami is a GREAT town. But having grown up there, I've been saturated. While I still love it, I no longer feel I'm missing out on anything by not being there, but it took a solid 25 years to get to that point
Good luck!