The bus to Burgos leaves directly from the airport, Terminal 4. You can get to Terminal 4 by taking the shuttle bus from your arrival terminal, one floor up from baggage claim, outside to the right. The journey between terminals takes about 15-20 minutes, depending on how long you have to wait for the next shuttle.
If you buy your bus ticket at the airport, you have to buy it inside at the Alsa kiosk, not on the bus. *The bus driver does not sell tickets.* The kiosk is located on the arrivals floor, and the buses depart outside, I think on the same floor, but it is well signed in any event. The bus to Burgos will leave from bus stands 1-4. The bus may or may not say 'Alsa' on the side, so check with the driver of every bus that leaves around your departure time to be sure your bus doesn't drive away without you (I've seen that almost happen many times!).
There is often a queue at the Alsa kiosk, so you may want to consider buying your tickets online, in advance. There is an English version of the Alsa website, and you can pay with PayPal.