I'm not an expert on the subject, since I've only been to Rio once, and was more interested in getting the feel of the city than getting to know the city itself (which is absolutely beautiful and chaotic, btw), but I had so much fun that I will probably do the same if I ever go back. Here's my tips:
Ipanema beach is not only a beautiful place to visit, is also a nice place to find a hotel, preferably near the posto 9 (that's a really famous spot, so you won't have problems finding it). Copacabana is much more touristic, and that, sadly, means more dangerous at night. Leblon beach is perfectly nice, but also posher, so it will probably be more expensive. Anyway, people from Rio still love Ipanema, so is very lively both by day and night. Near the posto 9 there are some nice and cheap bars (the fuller, the better, do what brazilians do). There are also nice shops behind the beach, so you can do lots of things without being too far from the hotel (something you'll be grateful for. It's a big city and public transport can be a bit tricky).
Santa Teresa and Lapa neighborhoods are so beautiful (mind you, quite derelict, so don't go there expecting to find colonial splendor), but taking the tram (bonde, they call it) to Santa Teresa and going back to the centre of the city is an absolute must. Views are spectacular from the bonde (including Maracanà stadium) and it's quite a shock to finish in the city center, a perfect example of the "ordem e progresso" futuristic grandeur (you will find there the cathedral, a sort of mayan pyramid featured in a Play Station add of some years ago). Lapa is also charming, and a good place to go out on friday night and have fun (carnival-like fun, streets full of people dancing and drinking to various sources of really high brazilian music. I love it, but also understand that is not for everyone).
The Botanical Garden is another thing I wouldn't miss. I am Mediterranean, and was absolutely flabbergasted by its tropical exuberance.
Never went to the Pao de Açucar or the Corcovado because it was cloudy and I had little time to spend in Rio, but those must be the places where you can really appreciate the great contrasts of the city: wild forest vs. incredible beaches, ordem e progresso vs. favelas, so I recommend you to visit at least one of them if you are luckier than me with the weather.
And last but not least, it would be mean not to recommend a school of samba rehearsal. There are twelve major samba schools in Rio and apparently they rehearse every weekend, so you can go to one of the rehearsals. I went to a Mangueira samba school final rehearsal before carnival and never had so much fun in my whole life (specially considering that they only play one song). The thing is that samba schools are normally in the heart of the favelas, so if you don't understand portuguese (the best way to decide what is dangerous and what is not is to ask people), befriend a cab driver (take always the same cab, I mean), who can also guide you safely to other parts of the city.
Anyway, if you understand portuguese, the portuguese wikipedia has lots of info about all these places. Enjoy in Rio!