You have to go to the péage of motorway A36. Take bus 34 in direction of Valentin from the bus stop Saint amourin the centre of Besançon until the stop Echangeur Valentin (it's almost the end of the bus route). Then walk in the direction of McD and you will see the péage. You should use a sign or ask people because cars are going both west and east from here. Note that there are no buses on Sundays so you should walk there, it's about 6 km.
From the center, you can walk to "rue de Vesoul", and there is a bus spot a little bit more after the station AVIA, next to a school. Using a sign with Valentin should be easy. It takes about 30 minutes walking from pont battant in the center.
You can also go to Dole by this way, but you can also go by the national road. Take the bus to Espace Chateau Farine, then you will have to walk along the road leading out of the city. Be careful as it's a 2-lane road where it may be difficult and dangerous to hitchhike.
Update 30 May 2015: It seems that there is no longer a bus 34. Instead you have to take bus 20 from the bus stop "Centre St Pierre" in the center of Besançon until the stop "Echangeur Valentin"
Southeast toward Pontarlier, Lausanne
The spot is at the petrol station, and never takes more than 30 min (can be 4...). For this you have to walk out of the centre to the crossing called Rivotte (near the tunnel under the citadel), then follow the road in direction of Pontarlier. Next good point is the round-about close to Valdahon, don't take a ride which goes closer than this point.
Same spot to get to Valentin from the center as described above. Wait for a long ride, or ask if people can stop you in "Cayenne", or next to Quenoche where cars have to slow down. Don't go to Rioz.
A local bus ticket costs EUR 1.30 and is valid for one hour on any buses. Be careful on Sundays there is no bus to go the péage!
You want to know where to camp? Where are the markets, have nice drinks or buy clothes in secondhand shops, go here, and share with us if you know more useful informations : MAP
For example, there is a place where locals are making free vegetarian meals for everyone each last sunday of the month. Check it out!