Rio was the capital of a Portuguese empire, then of an independent one, then of the republic, and it lost the title only in 1960 when Brasília took over. It never really got over it. The city founded in 1565 by Estácio de Sá still has the swagger of a capital: an opera house copied from Paris, a library in Manueline stone, a 1750 aqueduct, and between all of it granite peaks that rise straight out of the sea and make every street corner look like a postcard that somebody took a bit too seriously.
The honest pitch is this. Rio is two cities in one: the sightseeing city (Christ the Redeemer, Sugarloaf, the old center, Santa Teresa) and the beach city, where Copacabana and Ipanema are not scenery but the living room, the gym and the town square of the people who live behind them. Five days is about right for both. It's also a big, hot, hilly, unequal city with real safety rules, which this guide spells out instead of whispering.
The plan is organized by geography so you don't spend your trip in traffic. Day one is the old center, the port and the Monday samba at Pedra do Sal, day two is Christ the Redeemer, the Botanical Garden and Sugarloaf at sunset, day three is Copacabana, Ipanema and the Arpoador sunset, day four is Santa Teresa, the Selarón steps and Lapa, and day five is Tijuca Forest, Maracanã and a last night of music. Start on a Monday if you can, because the Monday roda at Pedra do Sal and the weekday-only Real Gabinete line up with day one. There is no day trip on purpose: the best ones are in the more-time section, and Rio itself deserves all five days. For the country around it, see Brazil.