It is a largely touristic city. While not so extensive as the capital Mexico City, it is still necessary to catch a bus out to the highways to hitchhike. Close to Oaxaca is the archaelogical site of Monte Albán. A little bit further away is Mitla, another archaeological site, as well as the natural site of Hierve el Agua.
North towards Puebla, Mexico City
Go to the central bus terminal (abastos), from there look for a bus that is heading for Etla, and ask them to drop you off at the entrance to the Highway 135D (they also call it la supercaretera), it shouldn't cost more than 10 or 15 pesos per person. From where the bus drops you you will have to walk about 5 minutes up the slip way. A little way down the highway there's a tree (for shade) and you can hitch from there. Traffic moves pretty quickly, so a sign is advisable. If you're having trouble getting a ride, try writing "caseta" (tollbooth) -- there's a tollbooth a few minutes down the road (driving), and pretty much every vehicle going by should be going at least that far.
South towards San Jose del Pacífico, Puerto Escondido
You can try hitching straight out of town on Avenida Símbolos Patrios. To get a head start, though, for around 10 pesos you can catch a shared taxi to San Bartolomo or, even better, to Vincente Guerrero. These (wine red coloured) share taxis can be found heading down Avenida símbolos patrios. From Vincente Guerrero, walk to the Y Griega (y junction) or ask the share taxi driver to drop you there. Traffic will be heading to Ocotlán, Miahuatlán, San José, and the coast. If traffic gets a bit thin after Miahuatlán, a shared pickup truck "taxi" from there to San Jose will cost you around 30 pesos