Hamburg Airport is 8.5km from the center of the city and is therefore pretty easy to get to with the use of the U-bahn. The S1 line connects the airport to the main train station (hauptbahnhof) in about 30 minutes. Tickets cost 2.60 eur (will cost 2.70 from the beginning of 2009), the trains travel between 4am and midnight, from then on a bus service (606) is put on between the train station and airport, this leaves every 30 minutes.
Tuifly, Germanwings and Flybe are the only budget airlines flying in and from this airport as far as I am aware. If you are flying to Hamburg with Ryanair, Wizzair or Jet2 then you will land at the distant Lubeck Airport which is occasionally labeled Hamburg when it comes to Ryanair sites! That one is 54km from Hamburg!
I found Hamburg airport to be quite a good place to hang out, as I arrived a little early - there is plenty of seating and the queues were not large at all - although my flight was not particularly busy! I also noticed that the prices at the shops and cafes were not ridiculously high as is quite often the case when it comes to airport shops.
A clean and organised airport, which is well connected to the city - I would easily use it again.