There are a number of public transportation in options in Rio de Janeiro, including city buses, executive buses (Frescao), the subway (metro), vans and even a Ferry service.  A seasoned traveler would consider Rio a very simple city to navigate, especially if you speak a little bit of Portuguese.   In general the public transportation options are very safe;  crime in the subway (Metro) is almost unheard of; the executive bus known as the “Frescao” and taxis are very safe.  However, taking the regular buses at night is not recommended for tourists. Given that Rio is a very cosmopolitan city, the transportation prices are very cheap.      

Taxi

Taxi system works in Rio de Janeiro quite good. There're rates which depends what time you get in taxi. From 9.pm till 9 am it's daytime night time tarrif as from 9am till 8 pm it's daytime tarrif.

Here you can find a taxi route cost TO & FROM including with traffic variances.

 http://www.taxiautofare.com/br/Defaul...

City Bus

City bus is a very inexpensive form of travel, tickets are R$2.75 for a non air-conditioned bus and between R$3,10 – R$3,50 for an air-conditioned bus.  In past, city buses suffered from criminals targeting the passengers, but now days they are safer, however it is still advisable to only take them during the day, and stay in the safer areas of the city.  This video from Geobits is very well made & helps to understand how to get a regular bus or Metro (see below) in Rio. This Wikipedia page has more specific bus information for Rio.

Here you can find how to get TO&FROM addresses by bus  with map and route including. http://www.vadeonibus.com.br/vdo/inde... 

Bus number and stations list.

http://www.rioonibus.com/rioonibus-wo...

The prices for an ordinary bus recently went up to R$ 2.75. If you have a CPF-number (tax number, also available for foreigners) you can buy a card which allows you to change the bus once within two hours without an extra charge. 

If you want to get by bus from Copacabana or Ipanema to the sights of Rio, you can have a look at this map.

Executive Bus (Frescao)

There is also another type of local bus called the Frescao. This bus runs on a route from Centro through Botafogo, Copacabana, Ipanema and to Leblon (and vice versa) neighborhoods.  It is more upscale/comfortable and is air conditioned and costs R$3.50. However, it is only available during the week.  The buses also run more frequently during the rush hours in the morning and evening.  Going in the direction of Centro , the bus can be flagged down on the beach road (Buses with plaques showing "Castelo").   Coming from Centro, the bus starts off from the Menezes Cortes garage and runs on a road parallel to the beach (but one or two blocks in). It runs on Barata Ribeiro through Copacabana, Prudente de Morais through Ipanema and Av. General San Martin through Leblon.

Well you do have the executive buses going everywhere in Rio de Janeiro and not just in the rush hours but all the time during week period. It is very easy and safer to get the executive buses than the regular one, especially if you are a tourist in the city. And the price does vary depending on the distance that it is going, for an exemple is not the same price to go Copacabana from the Downtown as to go to Barra da Tijuca or Recreio from Downtown.

Rio Subway (Metro)

Rio has a good subway service that serves a number of areas central to the Centro. There are three stops in Copacabana ("Cardeal Arcoverde", in the beginning; "Siqueira Campos", four blocks from the beach and about one third of the way down; and "Cantagalo", in the end of Copacabana, near Arpoador). At "Ipanema/General Osório" station, you can take special subway-buses to Leblon, Gavea and Barra da Tijuca.   At "Botafogo" station, you can take the subway-bus to Gavea, going through Jardim Botanico. In order to take these buses, you should mention, at the time of the subway ticket purchase, that you want to take the "Metro na superficie" and inform your final destination. The vendor will give you a ticket to the subway-bus along with your subway ticket, no cost added.  

Also, a good point worth making is that the Subway system is extremely clean, since it is privately owned, but with strong regulation.  Most stations are very new too.  Below a great photo of the Metro in Rio:

Van

Rio also has lots of vans which are generally privately owned. This is a very cheap option with many different routes. It is not advisable for tourists to use these as many are not licensed.  

Ferry Boat

The sister city to Rio and on the other side of Guanabara Bay is Niteroi. There are quite a few people who live in Niteroi and commute to Rio to work. There are several ferry services that operate between the Rio Centro (Praca Maua) and several points in Niteroi including the Centro and more toward the beach area past Niteroi (Charitas). There is a traditional boat as well as several "fast cat" hydrofoil boats. The price is about R$5.  Ferry boat is specially good option to visit the famous MAC  - Modern Art Museum at Niteroi.

Lonely Planet info about the ferry.

Gondola

Starting in 2011, a network of 6 gondolas will transport about 30,000 passengers from the hillside favelas into Rio.