Hamburg Airport has frequent flights to domestic and European cities with Lufthansa and most other major European carriers. Low-cost carriers include Ryan Air, Air Berlin, EasyJet and Eurowings. There are also a handful of intercontinental flights, such as to New York and Dubai.
The S1 S-Bahn connects the airport directly with the city centre, including the Hauptbahnhof. The journey takes 25 minutes and costs €3.30. Taxis cost €20 to €30, depending on where you're going, although it's more likely to be the latter for the city centre; the trip into town should be around 30 minutes, longer during peak hour.