Thanks VidaNaPraia! We are going to Sugarloaf, Corcobado and Santa Teresa on the 2 and 3.

I agree with VidaNaPraia, just relax, people watch, and hang out around Copacabana since that is where you will want to be for the New Year's Eve festivities. Better to not get flustered running around on your first day.

Just a couple of other suggestions for following days, if you have time.
You can take public transit (or taxi) to the Jardim Botânico, which is next to Lagoa Rodrigo de Freitas. The garden is an oasis of calm, and you can often see toucans and monkeys high in the trees. You can also take some interesting, as-seen-from-below, photos of Corcovado from Lagoa. On the way back to your hotel, you can stop for a drink in Ipanema, the neighborhood adjacent to Copacabana, which is, in my opinion, less hectic, more stylish. This neighborhood is also culturally significant: it inspired the celebrated Bossa Nova hit, Girl from Ipanema (Garota de Ipanema).
http://en.wikipedia.org/wiki/Rio_de_Janeiro_Botanical_Garden
Video of Girl from Ipanema (ignore the narrator's pronunciation of Brazilian portuguese):
http://live.wsj.com/video/the-girl-from-ipanema-turns-50/FA2D5718-6F37-4BE4-B205-64E1C9048E3C.html#!FA2D5718-6F37-4BE4-B205-64E1C9048E3C

Just a bit farther away, in the opposite direction, is the city of Niteroi. You can take public transit from Copacabana to Niteroi's stellar attraction, the Niterói Contemporary Art Museum (Museu de Arte Contemporânea de Niterói — MAC), which was designed by Brazil's preeminent 20th-century architect Oscar Niemeyer (who died at the age of 104 earlier this month). The primary reason to visit the museum is to see the building itself and to admire the across-the-water views of Rio de Janeiro. The art collection is not of great significance when compared to Niemeyer's building.

The museum is not very large or imposing, but unusual, in that it is shaped like a flying saucer! While Niemeyer was highly original, the building also seems influenced by the architecture of Frank Lloyd Wright.
http://en.wikipedia.org/wiki/Niterói_Contemporary_Art_Museum