The journey from La Paz to Copacabana takes three to three and a half hours. In general bus travel in Bolivia costs 1 USD per hour, so expect to pay 3-4 dollars. More if you prefer a more luxurious bus. Buses leave from La Paz’s cemetery bus terminal. The majority of the buses leave at 8:00 am. During the bus ride there is a stop where you will need to take a ferry across Tiquina Strait to meet the bus, which will also be ferried, on the other side. The bus will arrive in Copacabana at Plaza 2 de Febrero. Buses leave from the city at Plaza Sucre.
Also note that it may be best to take larger tourist buses on this journey, since they tend to be safer. In rare instances there have been reports of kidnapping and robbery on smaller minibuses. This is by all means not always the case, but just be aware of the possibility, and choose your bus company carefully.