Busan is the second biggest city in South Korea and probably one of the best places to hang out in the country
North towards Daegu, Seoul
You can hitchhike north from this tollgate. There are a few busses going to the spot. The staff there even helped me make a sign.
Update: Two years later in July 2019, User:Hippietrail tried this spot for an hour and a half on the way out of the tollgate with no glimmer of luck. The left lane is heading to the highway to Daegu and cars in this lane would have to cross several lanes in a short distance to pull over, and there is no unused space for a car to pull over safely. Perhaps it works better at very quiet times. Or perhaps walking through the tollgate toward the tunnel to stop cars before they get to the tollgate. Or perhaps being right at the gate where people have to reach out of the window is the only way?
Update (november 2019) : I tried another spot from Yangsan Service Area (35.323198, 129.056064). It is easy to get there from Busan city center by public transport : Subway Line 1 --> Bus #16 (7 stops) --> 5 minute-walk to enter the station with a small trail. Once at the station, you will easily find a ride towards Daegu and Seoul. It took me about 15 minutes. Enjoy!
West towards Gwangju, Jinju and the rest of southern part of Korea
There are several highways leading westwards out of Busan. Probably the easiest one to reach and hitch is the number 10 or the Namhae expressway. There is a big tollgate called BukBusan. You can reach it by taking the brown (third) subway line all the way to the last stop at Daejeo. Leave the station on the exit 1 and turn right. Follow the big road going under the light railway line (under construction) for about 1-1.5km (just count two stations above you on the light railway line) and when you see a church on your right, pass it and turn right in to the small street yourself (there is one another church along the way too, don't get confused). When you turn right on the church, there will be a few more quick zigzag turns and you come under the highway, so just find your way up and the tollgate is right in front of you. Make a sign of where you want to go (in Hangeul) and GOOD LUCK :)