There is no professional sport in Copacabana at all; what people do here is walk uphill at altitude, and the lake is too cold for anything long without a wetsuit.
Nearest top-flight football: La Paz
Copacabana has no professional club. The nearest are Bolívar and The Strongest, both in the Bolivian first division and both playing at the Estadio Hernando Siles in La Paz at 3,637m — three and a half hours away by bus, and famously hard on visiting sides.
Cerro Calvario
The hill above the town, climbed on a stepped path past the stations of the cross. Thirty to forty-five minutes up if you are acclimatised and considerably longer if you are not; locals walk it in the evening for the light on the bay.
Copacabana to Yampupata
About 17km along the shore road and lakeside paths to the tip of the peninsula, four to five hours and mostly flat, with boats at the far end across to Isla del Sol. The classic day on foot here, and you can bail out onto a passing minibus.
Walking on Isla del Sol
The southern end around Yumani is straightforward and reliably open — the ridge path, the Inca steps and the spring. Access to the northern communities has been disrupted by disputes between the island's villages in recent years, so ask in Copacabana before buying a through ticket.
Acclimatise before you exert
At 3,815m the first two days are for walking slowly, not for running or long swims, and arriving from La Paz rather than from the coast helps a great deal. Titicaca stays cold year-round, so open water here means a wetsuit and short distances.