Oaxaca's capital is known in
Mexico as
Oaxaca tout court, although its complete name is
Oaxaca de Juarez (for
Benito Juarez: he was born in
Gueletao, a small village not far from the capital).
The state of Oaxaca is one of the poorest in Mexico: the contrast between the reasonably developed capital city, the burgeoning tourist areas on the coast and the incredibly poor mountain areas is very strong.
The turist usually sees the seaside resorts and "hippy" areas, like Ixtapa, Zihuanatejo and Puerto Escondido, and the areas around the capital. By the way, if you visit the artisan villages close the capital (like Teotitlan del Valle) keep in mind that these are the richer villages.
The state of Oaxaca is famous for its food, and rightly so. In Oaxaca City, go to the Mercado Benito Juarez and eat in any comedor. What's to eat in Oaxaca ? Mole rules, there is a special kind of tamale (wrapped in banana leaves), tlayudas (or clayudas), chocolate, and many fine mole-like sauces like amarillo, colorado, pipian. Quesillo or Queso de Oaxaca, the kind that comes in huge balls and makes strings when it melts is also very good here.
Notice that in Oaxaca an empanada is actually a giant quesadilla with quesillo and flor de calabaza. Very very tasty.
There are also many fine crafts, like rugs, Oaxaca black pottery, alebrijes, woven goods and knives. No less than three knife-makers operate in Oaxaca.
Oaxaca is also famous for its many archeological sites: the most famous are probably Monte Alban and Mitla, although I have found that Yagul and Dainzú are very interesting as well. Monte Alban also has a rather fine museum, and it is easily reached from Oaxaca City.