There are frequent buses from Quito. Buses usually depart from Terminal Norte Terrestre de Carcelén, there are different options which will imply terminal fees. (10/2018) Information on bus schedules is available at andestransit.
Most buses make their final stop in Tulcan's bus station. It's small, but there are taxis and restaurants around and open even on Saturday nights. Hotel/Restaurant Acacias is across the street.
Colectivos, at parque Ayora, take you to the border for 75 cents (You can go to the parque Ayora for 30 cents at the bus stop on the main road above the terminal). There are also city bus service between bus terminal and city center. Unfortunately the taxis have fixed the price to and from the border into town at $3.5/trip, even though this is more than it should be (or would be if they had meters).
There's a place called Hotel Las Acacias just across the street from the bus stop which has clean comfy rooms with hostel prices and you save money by not having to get a taxi to another place. There's WIFI, but only in reception or in rooms close to it, we stayed in the room directly above and had a good WIFI signal, but also had constant noise and light from the street. Good for a stopover before doing the crossing and journey to Popoyan/Bogota/Medellin in the morning which is apparently much safer than doing it at night as that particular stretch of road is "dodgy." There are also another couple of places to stay right next to Acacias, but I didn't stay in them so can't give an opinion...
Ipiales is just on the other side of the border and can be reached with collectivos or taxis. Colectivos depart from the bus terminal and from Parque Ayora in downtown.
Take a taxi to Rumichaca bridge ($3, 8/2014) or colectivo (75 cents, 10/2018). Walk across the bridge, taking care of formalities with Ecuadorian and Colombian officials. On the Colombian side, take a colectivo to the Ipiales downtown/bus terminal for 2,000 COP (or $1 if you don't have pesos).
Another alternative is to take a Quito-Cali shuttle which makes a stop in Tulcán. This service will help you navigate the border formalities .