Rio de Janeiro City
General information: City Tour - Rio de Janeiro
In the opinion of a foreigner (Norwegian) who have completed course as a licensed tour guide for Rio de Janeiro and lived here for years including in center region: "It is quite possible to stay for weeks in Rio without visiting downtown, unless you have particular interest in buildings as there is not much to see there"
Even though center is not the most beautiful part of Rio de Janeiro it does have interesting elements primarily due to its history including as a capital of Brazil.
When it comes to safety in downtown there are no substitutes for common sense.
I will give some advice, but there are no substitute for common sense and prudence.
City of Rio de Janeiro should be visited during daytime and weekdays.
Generally there are not many walking tours provided by tour guides to downtown during sundays due to safety reasons.
Meaning that if you intend to walk in downtown during Sundays or after nightfall you should know where to go and not to go.
In streets with movement during daytime, the risk of getting robbed is very low, even as a Gringo.
If you take metro to Cinelandia to enjoy buildings in the square or decide to visit Lapa via street Rua do Passeio when there are many people there (which it usually is during weekdays and Saturday) the risk should, as far as I know, not be too big.
However, try to avoid taking shortcuts by empty streets after nightfall or in weekends.
Example, do NOT take shortcut via R. Evaristo da Veiga instead of using the above mentioned Rua do Passeio.
In same manner - a crowded Rio Branco is much safer than Avenida Paraguai (which should be avoided when empty).
I will try to here develop a directory of the most interesting places to visit in downtown Rio de Janeiro.
Places to visit in Downtown RioAvenida Republica do Chile
São Bento Monastery