Bariloche (officially San Carlos de Bariloche) is a city in Rio Negro, Argentina.
North towards San Martin de los Andes/Villa La Angostura/Chile
You can take the 71 bus for 12 pesos (~1 USD, as of December 2014) to Dina Huapi (about 15km north of Bariloche). Stay on the bus until it turns off the main road. You can ask the driver, but it's pretty obvious, and there's a stop as soon as it turns. Depending on which 71 you're on, you'll either have to walk about 100m or 1km along the road to a police checkpoint along a curve where traffic slows considerably and people can pick you up. Another way is a traffic light right next to the bus terminal within walking distance of the town center (3 km). You can also always start hitching on the main road inside the city which seems to have a lot of tradition. Chances are good you will be picked up before the bus even arrives. To continue on RN40, you need to take the turnoff at Villa La Angostura. If you want to get to Chile, continue with traffic on RN231; check details in the Villa La Angostura article. An alternative route to San Martin de los Andes could be via Villa Traful on RP65. Herefore, continue on RN237 instead of hopping off at Dina Huapi and take the turnoff at the ACA petrol station when Rio Traful hits in Rio Limay.
South towards El Bolson/El Calafate
The way out of Bariloche towards El Bolsón is through the barrio 2 de abril, which is rather universally described as a dangerous place and lots of people recommend avoiding it for your own safety and because drivers are also less likely to stop there, due to its reputation. Hostel owners speak of a few cases per year where people get a ride but are then immediately forced to get out of the car and eave all their stuff with a gun held to their face.
Looking for public transport to get out of town can go a long way. There is usually a fair amount of hitchhikers on some spots on the stretch of RN40 by Villa Lago Gutiérrez, trying to hitch a ride to El Bolsón. Christianmn had to wait for a few hours until finally getting picked up, together with a young french couple also hitchhiking in the same place, by a truck driver who was heading to Esquel. However, despite that spot being not highly trafficked (actually, most of the cars were turning at the nearest crossing on the way to Villa Lago Gutierrez), he saw that other hitchhiking groups and couples had better luck and found a driver in approximately less than an hour or even half an hour.
West towards Llao Llao/Col. Suiza
RP77/RP79 can take you on a round trip to beautiful Llao Llao and Col. Suiza. There are suburbs for miles leaving this direction, so it might be more reasonable to take a bus to exit the city. Cerveceria Patagonia offers you incredible views over the entire National Park at expensive food and drink prices. Part of the northern circuito chico can be walked within beautiful forests.