Copacabana is about six streets and a bay, so nothing is far; the choice is between the noisy lakefront strip, the plaza beside the shrine, and the quieter slopes where you trade a climb for the view.
Avenida 6 de Agosto — the traveller strip
The street running from the plaza down to the water, lined with hostels, agencies, money changers and trout restaurants. Everything is two minutes away, and it is the loudest address in town at a weekend.
Around Plaza 2 de Febrero and the Basilica
The square in front of the white shrine, where the pilgrim buses, the processions and the vehicle blessings all happen. As central as it gets, and correspondingly noisy on feast days and Sundays.
The Cerro Calvario slope
The lanes climbing north-west towards the Calvario hill, where guesthouses have the best angle on the bay and the sunset. The price is walking back up to them at 3,815m with a pack on.
Isla del Sol, Yumani — practical tip
Boats take an hour and a half or more each way, which makes a day trip mostly boat. Sleeping a night in Yumani is cheap and gets you the island without the crowd, but there are no vehicles, limited electricity and no ATMs — take cash and a torch.