Valka-Valga border crossing
The Valka-Valga border crossing is a point along the Estonia–Latvia border. Valka is the name of the town on the south (Latvian) side of the border, while the town on the north (Estonian) side is Valga. It lies on the European road E264, Latvia national road 3 and Estonian national roads 3 and 6.
This is really not really much of a border crossing at all: there is only a sign reading "Estonia" if driving north or "Latvia" if driving south, but there are no checkpoint buildings and nowhere to stop along the side of the road. Your driver will leave you at one of the intersections before or after the border.
North towards Tartu
If you want to hitchhike to Tartu, don't hitchhike from near the border because most of the traffic is local. Instead walk about a kilometre along the road signposted "Tartu" until you get to a Lukoil petrol station and an intersection marked "Tartu" to the left and "Kesklinn" (Valga city centre) to the right. Hitchhike from just after the pedestrian crossing at the intersection, and you'll be able to stop a car leaving Valga and going long-distance.