Hi there,
I'm looking for a safe, reliable bus company that does the Cusco to Puno route and also a bus company that does the Puno to Copacabana route.
Thanks

Not sure of the bus we booked in Cusco, but it was just through one of the many travel agents. All went fine, stopped at some Inca sites for half an hour or so along the way, stopped at a reasonable restaurant for a buffet lunch, all good stuff.
Then in Puno, booked Copacabana bus through hotel, which included taxi transfer to bus station and reserved seats, but in our case taxi-man quite adamant that we had to pay him after the taxi trip and ended up in seats on opposite ends bus ( all just part of the joys of travelling).
At the Bolivian border, don’t listen to bus-man about changing your currency at a place near the border, he must be on huge commission as exchange rate very bad, much better at any of the many street money exchangers in Copacabana (ATM's in Puno and La Paz, but none in Copacabana)