I second Lübeck, although the town is a good 10-minute walk from the station.
For railway interest, arm yourself with a local Lander-ticket and have a wander! There are loco hauled regional services to Lübeck / Travemunde Strand, Kiel and Flensburg with DB, all double deck stock hauled mainly by ex-DR class 112s and 143s. Metronom run to Cuxhaven with new-ish class 246 diesels and to Bremen with class 146 electrics. Nord-Ostsee Bahn (NOB) use new class 245s out of Hamburg Altona on the semi-fasts towards Westerland.
Inter City services also run to Westerland - there are four each way which are electric (class 101) as far as Itzehoe, where pairs of class 218 diesels take over. As also mentioned above there is one train pair which uses 218s to/from Hamburg Hbf. The Westerland line is flat but has some interest - there is a large bridge over the Kiel canal at Hochdonn, and the final leg from Niebull to Westerland crosses the Hindenburg Dam (google 'Marschbahn' for further information). Niebull to Westerland is also served by Motorail services, to which a class 628 DMU is added at Niebull - good for novelty value!
There is also the huge bridge at Rendsburg as mentioned above, which is well worth a look. For a trip out, you can cross over it on a Flensburg service, alight at Schleswig and take a DMU to Husum (nice little town with a harbour) then back with NOB over the Hochdonn bridge into Hamburg. You can do that with a local Lander-ticket as long as you avoid the ICs.
Other IC services are mostly in the hands of 101s, with the odd 120. There is also a 2-hourly RE service to Rostock, which serves Schwerin, worth a look as it is a pleasant town. The ubiquitous ICE is in plentiful supply as you would expect.
Finally, Miniatur Wunderland, the huge model railway exhibition, is down in the trendy Hafen area.