The flight connection decides where you meet immigration
Xoxocotlán International Airport, code OAX, serves Oaxaca from Santa Cruz Xoxocotlán, a southern suburb of the city. It is a small airport, with much of its flying coming from Mexico City. From the United States, you can also reach it nonstop from Dallas/Fort Worth, Houston and Los Angeles.
For a traveler starting elsewhere in the States, a connection through Mexico City is a normal way to reach Oaxaca. That stop is not just an aircraft change. You clear Mexican immigration and customs where you enter the country, so a connection through Mexico City puts those formalities there rather than at OAX.
I’d weigh a nonstop against the Mexico City connection with that border step in mind: on the connection, immigration and customs sit between your two flights. The arrival airport may be small, but the itinerary can still have an awkward middle.
There is also a first-class bus from Mexico City’s TAPO terminal to Oaxaca for travelers building the trip overland. It arrives at the first-class bus station north of the city center; tickets are available online and at the station.
At OAX, the counter sells the ride
After arrivals, look for the airport transport agencies inside the terminal. Show the staff your destination address. They assign the central zone if you’re staying in the historic center, while neighborhoods farther out fall into the outer zone. The payment depends on that classification, not a conversation with the driver at the curb.

You can pay at the booth with cash or a card. Take the receipt outside and hand it to the driver. Do not pay the driver again. The paper has already done the financial part of the journey; the driver supplies the wheels.
Ordinary city taxis do not collect arriving passengers at the airport terminal. Authorized airport transport does.
Shared van or private car: decide what you are buying
The colectivo, a shared airport van, carries passengers to their individual addresses. It leaves when it fills, and the order of drop-offs depends on who else booked a seat. You might spend some time waiting before the vehicle moves, then ride through other passengers’ drop-offs on the way to yours.
A taxi especial, a private car for your party, does not depend on filling seats with strangers. The catch is availability: a private car may not always be waiting when you arrive. The desk can tell you what it can sell at that point.

If I were traveling alone without a pressing arrival plan, I’d take the shared van. If the flight landed late or I were arriving with companions, I’d ask about the private car instead. With a group, the private option is not a luxury argument; it is a question of how many people are splitting the same vehicle.
The return ride starts in town
The airport transport company keeps an office near Alameda de León, by the cathedral in central Oaxaca. That is where you buy a shared-van seat back to OAX.

Bring your flight details. The office uses them to set a pickup that gets you to the terminal in time.
The van collects you from the place where you’re staying. I’d visit the office a day ahead instead of leaving a return booking as a loose end on departure morning.
Buying the ride in at OAX doesn’t arrange the ride out.
Your passport and the stay you are actually granted
US citizens do not need a visa for a tourist visit of up to 180 days, but that upper limit is not a promise about your own admission. The Mexican immigration officer determines how many days you receive when you enter.
If you fly to Mexico, bring a passport book. A US passport card is not the document for boarding the flight.
For air arrivals the immigration record, the FMM, is digital, and the length of your authorized stay is marked when you enter. Look at that mark before you leave the officer’s desk. On this trip, the desks make the decisions.