Miniatur Wunderland is great - I'd definitely recommend booking in advance, however, as it gets really busy.
I always tell Brits to go and see the Beatles stuff on the Reeperbahn when in Hamburg, but maybe not for a 19 year old! 😂The Speicherstadt is cool for a walk around, and definitely have a look at the Elbphilharmonie concert hall. If you walk through the Elbtunnel from the city, you can get Instagram-worthy shots of it.
The Schanzenviertel is a cool part of the city for younger people. Lots of students hang out there and there are independent shops, restaurants and bars - all very quirky.
Leftfield suggestion . . . even if you're not a fan of seafood, the fish market is quite an experience to witness. There's a good atmosphere, always packed full of people and very traditional for Hamburg.
Oh and the boat tours run at night, too, so if your days are quite busy, then that's maybe a good option for you. The city looks very beautiful lit up.
Have fun and any questions, I'd be happy to help. (I'll stop now before I get too homesick 😭)