The Tulcán airport has been closed for several years.
All formalities are taken care of at the Rumichaca crossing (6am to midnight daily), 6km from Tulcán. Even day-trippers to Ipiales will need their passport stamped.
Minibuses to the border ($0.75, shared) leave as soon as they are full from the corner of Venezuela and Bolívar on Tulcán’s Parque Ayora (near the bandshell). Colectivo axis ($0.85) leave from across the park, in front of the Sana Sana pharmacy (the side of the park closest to the Bolivar statue/fountain).
On the Colombian side, entrance formalities are straightforward. Check with the Colombian consulate to make sure your nationality doesn’t require a visa. Visas are good for 30 to 90 days.
From the border, it's easy to catch a taxi ($1) to Ipiales, the first town in Colombia, 2km away, where there are plenty of hotels and onward bus connections.
If you want to cross the border at Los Chiles, you can take one of the white-and-green camionetas at the corner of Garcia Moreno and Arellano. These are cargo vehicles but will take up to two passengers in the front ($1, 20 minutes).
At the time of research (June 2018), hundreds and sometimes thousands of Venezuelan refugees were crossing daily into Ecuador at Tulcan. This caused some travelers we talked to delays of several hours. Be aware of this situation if you plan to cross here from Colombia.
Buses traveling to Ibarra ($3, 2½ hours), El Ángel ($2, 1½ hours) and Quito ($6, five hours) leave from the bus terminal around 2km southwest of downtown on the road to Ibarra. Note that there can be a customs/immigration check between Tulcán and Ibarra, which in our research time delayed the bus by 30 minutes.
The Quito bus does not go to the Ibarra terminal, but stops on the outskirts of town. If you want to go to Otavalo, you can ask the driver to let you off on the Panamericana there for $1 extra (the best place to get off is at the cemetery).