No matter which direction you're headed, you can catch a shared taxi/van in direction of Bacalar from around the intersection of Aguilar and Independencia downtown. It's 20 M$ to Bacalar, for 10 M$ you can go as far as the place where the Cancún road and the Escarcega road intersect, generally known as el puente. If you're going to Belize, you could probably be dropped off at the turn-off to Subteniente López for 5 M$.
North towards Cancún
There's a bus stop right after the bridge mentioned above. Traffic is going fast and people aren't extremely eager to stop, but you'll get a ride at least as far as Bacalar. If you don't mind the ten extra pesos, you could probably save some time by taking the van until Bacalar right away and starting to hitch from there. Alternatively, you could also start hitching from the topes in Huay Pix.
West towards Escarcega
The van mentioned above turns north at the bridge, and there aren't any really good spots right before the bridge although there might be right after. So if you want to be sure to have a good spot to start, try the topes in Huay Pix.
Accommodation and Sleep
The Chetumal fire station isn't open to hosting travellers anymore as of March 2018, unfortunately it seems that they have had bad experiences.