Kayseri is a city in central Turkey.
North, West, and South towards Everywhere not East
Take a tram from the main square next to Kale (Fortress) in the direction of West and get off at the stop called DSI. It is east to notice because it just after the tram turns to the left from the main road. Get back to the main road and hitchhike behind the crossing. There are traffic lights so cars will go slow. From there ask for a ride at least to Boğazköprü, off to the west of Kayseri, where is a big roundabout from where you can hitchhike all directions should the car you are in be not going already where you are going.
There are Dolmus going to Boğazköprü roundabout too. I saw few of them, for instance those going to Boğazliyan. You can ask for a ride to this roundabout. Probably it is easier than to hitchhike of the city center. They pass through the main road going off the city and you can stop them pretty much everywhere.
If you are hitchhiking in the direction of Yozgat use this method too. This road is much busier than the road directly from the city to the North.
The train station is open all night. There are plenty of benches to choose from, it is warm and you shouldn't get chucked out. The loudspeaker announcements are extremely loud, however, so use earplugs.