Hi Jennifer. Rio has an expansive subway system (http://www.metrorio.com.br/en/)Rio subway, but you can also take cabs, and, depending on what areas you want to visit, cab may be faster. If you're a first time visitor, you may want to stay along the beach in
Ipanema Beach; Leblon is also on the beach, but is the scene of heavy protesting right now. There are dozens of bars and restaurants in the hotels and lining the beach, so you won't want for any ideas, and it's safe since it's often crowded until the late hours.
If you've visited before,
Santa Teresa is an artsy neighborhood on top of a hill, but definitely more adventurous than the beachside nabes. Also, the
Lapa - Rua do Lavrádio district is full of nightlife. I really like
Rio Scenarium but
Lapa 40 Graus is popular, too. Lapa is a safe neighborhood that can be plagued by pickpockets during the night, so just be smart about your personal items.
If you decide to go by taxi, try
Transcoopass Rádio Táxi at the airport, and get their card before you disembark so that you can call them from wherever you are. They have treated me best on my travels in Brazil thus far.
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Jeniffer, I live here in Rio. If I were a traveller I would go to Ipanema Beach (which an urban beach, but much better than Copacabana, which is too touristy). I also love Praia da Reserva and Prainha beaches, which are a little bit far away from the center but really really beautiful (it's a little bit farther than Barra da Tijuca).
Cultural centers are also great to visit. You should know the Museu de Arte Moderna do Rio de Janeiro, Centro Cultural Banco do Brasil (CCBB) (also known as CCBB), go to Lapa - Rua do Lavrádio, walk around Arcos Lapa and the pictoresque Santa Teresa.
Oh and don't worry too much about violence. Really... it's not that bad.
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Q: What's the best to way to spend 3 days in Rio?
A: It will depend on what you enjoy. In Rio there are many tourism options such as beaches, sights, bars, clubs, extreme sports, night parties, samba schools. etc..
Q: How can I safely travel within the city?
A: To safely travel within the city I strongly suggest you to use public transports, such as bus and subway, and avoid alternative transports, because some of them are not legal, and so, not safe. The cabs are good choice too.
Q: Where is a good night time hangout that is safe?
A: There are many options at the Downtown and at the south of the city but, I think that the most requested is Lapa Arches and around.
I hope you enjoy Rio.