There are several public transportation options in Rio that you may wish to explore. All of the options are relatively inexpensive and generally very reliable -- however it does require a little research before you set off. Taxis are by far the safest option for tourists who are visiting Rio for the first time and the small premium you pay for their convenience is generally well worth the expense.
The subway system (Metrô) is very good and easy to understand and will get you to most of the popular spots in the south zone. But the subway train alone may not get you to everywhere you want to travel. The subway when combined with a "special metro or express" bus (purchased together) is an enhanced option that gets you even closer to many of the popular destinations.
The city bus lines are definitely the least expensive option to travel throughout the city. However, there are many different bus lines and they generally don't run on a set time schedule. If you know a bit of Portuguese and are familiar with your destination, the bus could be a very good option. Vans are the newest (fast and cheap) option and are similar to buses in both routes and cost – however, many are unregulated and you need to exercise caution.
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"I have found out that there ain't no surer way to find out whether you like people or hate them than to travel with them" - Twain