Curitiba is a city in Brazil.
North towards São Paulo
Take a bus from the center in the direction Centro Politécnico. From there, go to the university campus (last station). The campus is next to the highway BR-116. Quite a lot of drivers are heading to São Paulo.
Alternatively, take a bus at 6am from Terminal Guadeloupe to Jardim Paulista and from there to Capivari Lake (at a stop outside JP Terminal, ask around). You will come along several big service stations full of trucks, the last one right by the lake. Apparently there are only two connections daily. (2017)
The easiest option. Take the bus in the center which goes to the terminal Barrio Alto. Then in same terminal without paying additional price take the bus 341 или 342 to go until Estrada da Graciosa. There is a traffic light with enough space to hitchhike. Also there is a gas station (in 1km by walking).
South towards Florianópolis
Take a bus from the center to the bus station Pinheirinho. The bus station is pretty close to the highway BR-116. Another option is to take the bus that goes to the airport (Aeroporto) in a tube station Rodoferroviária. From the airport head to BR-376 which goes to Florianópolis.
West towards Foz do Iguaçu
Take a bus towards the bus station "Campina do Siqueira". From there you walk towards "Parque Barigui", it's a 15 minutes walk in a straight line, and that road leads to the countryside of Parana, including Foz do Iguaçu (Iguazu Falls), Maringa, Londrina, Guarapuava, Cascavel, Castro (Canyon Guartela, 4th bigger canion in the world) and also towards Mato Grosso do Sul (Pantanal)
East towards Ilha do Mel
Go to the bus station "Guadalupe". From there you can take the bus that goes to Posto Paris for around R$5,00. Be careful because Posto Paris is the name of the bus line, but it is also the name of the Gas Station that you want to go, so let people know that you really want to go to the gas station and they will tell you where to get off. The bus goes to another nearby town called São José dos Pinhais through the highway. You only need to get off nearby some big Renault shop, from there just continue walking for another 15 minutes along the highway till you reach the Gas Station. Alex and Bin got a ride there to Pontal do Paraná and than the boat to Ilha do Mel, which costs around R$30,00.
Places to avoid
Avoid walking at night in places far away from the center. Don't talk with suspicious people and don't let them know you are a foreigner.
Bus tickets cost R$ 3,70 in Curitiba. You can plan your trip inside the city using the website onibuscuritiba.org.