All the options for getting into Budapest from the Airport are to be found here. Rather than me rehash the options you can read them at the airport website and decide:
Taxi Fare to Pest will run you about 7,000 ft or something less than $30; a bargain as international airports go. Ignore those hawking cab services when you enter the arrival hall and head to the nearest outside door. You will see a kiosk on your left for FoTaxi which has the airport contract. They are fair and decent. Actually very good service. Write Nyugati pályaudvar on a slip of paper and show it to one of the clerks in the kiosk and they will make sure the tax knows exactly where to go.
The majority of the trains to Debrecen leave out of the Nyugati pályaudvar (Western Station), which by coincidence is the same station that is connected in a roundabout fashion to the airport. Still, if you took a taxi you would probably cut an hour and a lot of trouble in the connection. You can see the schedules here:
More often than not I end up on a 6:30am return flight. It is miserable. Our accommodations are very near the opera house, and I have the taxi pick us up at 4:00 am. That allows a half hour to get to the airport and two hours to check in, clear security, etc. The Budapest airport is pretty nice but it suffers from a little disorganization at the check-in counter and my showing up a little early means we beat the worst of the mayhem. To return to the airport and any other taxi service other than the airport into Budapest I like City Taxi a good reliable company with clean cars and decent drivers:
+36- 1 -2-111-111
Where to stay? Well, you said you might want to see some of Budapest; I guess the day before your flight out. In that case since getting to the airport is only a 30 minute issue I would suggest you stay in town and not near the airport. If you have just the one night then stay at the Intercontinental or the Marriot on the river front. If you have three nights then stay deeper in Pest, maybe the K&K Opera.