Flying from Cuzco
The closest airport to Copacabana is Juliaca, Peru. The only two airlines that fly into Juliaca are Avianca and LAN, though keep in mind that foreigners pay two to three times the prices that Peruvians pay for the same ticket. If you fly to Juliaca, click here to learn how to go from the Juliaca airport to Copacabana by land. I think it's actually more of a hassle than taking a direct bus from Cuzco to Copacabana. Not to mention ten times the price.
Taking a bus from Cuzco
Go to the main bus terminal. Careful because Cuzco has a few different bus terminals. To be sure you're going to the right one, tell the driver "Al terminal terrestre." If you want to be sure you're going to the right place, add "Para ir a Copacabana, Bolivia." Near the entrance to the terminal, you'll find lots of kiosks offering tickets to Copacabana. Titicaca bus lines is one of my favorites, though they're pretty much the same to me, to be honest. Prices are usually around $20 for a direct trip that lasts about 12 hours.
Travel agencies in Cuzco also sell bus tickets to Copacabana, but they will charge you two to three times as much - and you'll still have to go to the bus terminal to catch your bus.
I suggest buying your ticket in advance. When I'm in Cuzco, I head to the bus terminal in the morning to buy my ticket, with plans to leave the same night.
Travel agencies in Cuzco also sell bus tickets to Copacabana, but they will charge you two to three times as much - and you'll still have to go to the bus terminal to catch your bus.
I suggest buying your ticket in advance. When I'm in Cuzco, I head to the bus terminal in the morning to buy my ticket, with plans to leave the same night.