I recall that the topic of day trips from Hamburg has come up here quite recently - probably buried in a thread about things to do IN Hamburg. I think that suggestions included Lübeck and Lüneburg - both very good suggestions, and both easy trips by public transport.

If you do a search you should hopefully find these recent threads!

As for Denmark... somewhere like Tĝnder (just north of the border) is probably just about do-able as day trip, but he'd be spending most of the day on the train (it's about three hours each way)