S-Bahn is short for Schnell-Bahn which are the suburban trains. In German cities you have the following: U-Bahn (big blue U signs) - Untererdisches Bahn so Underground, normally a small network in the city centre S-Bahn (green S signs) - Schnellbahn so suburban trains, normally still within the city limits, part tunnels part not, looks like a proper train and not like the DLR Strassenbahn/Tram - is surprisingly enough a tram Proper long distance trains go from Hamburg Altona and Hamburg Hauptbahnhof (Hbf for short) you can buy day cards or a Streifenkarte where you just stamp off the number of stripes that you need. It is usually 2 for an average journey. You can buy them all from vending machines in the stations or on the bus/tram NOT in the trains though Not having a ticket will cost you about 30 quid each - the inspectors all speak English and you can spot them by their belts (like a holster).
A final tip: avoid using the stairs in stations and always use the escalators even if they are broken - this is because the satirwells (at least in Frankfurt, since then I've not tried elsewhere) are often used by druggies/alkies, Hamburg has been cleaned up a lot though so don't let this put you off.
You'll get route maps off of the Hamburg tourist site but as you are so central you might actually not need to use a bus while you are there Of course if you get fed up with the whole thing you could get the Eurocity train to Copenhagen because that isn't that far! You are flying to Hamburg itself right? Not with Ryanair? Because Ryanair flies to Lbeck and that is about an hour away from Hamburg... and then you'd have to get the bus.
Seems I originally had a board here just to gather some content for Andy's stag do to Hamburg, Germany.