I’m a longtime lurker who registered just to answer this question. 
I have family in Hamburg, so my experience is colored by that – the first time I ever stayed at a hotel in Hamburg was a couple years ago – but hopefully a little helpful.
Where to stay: For hotels, any place that meets your quality/price requirements around the Alster should serve you best. We stayed at the Hamburg Marriott, and that was a perfectly fine hotel tucked away. The Fairmont Vier Jahreszeiten is a famous hotel, as is the Hotel Atlantic Kempinski, but very expensive, and I can’t give you a perspective on whether they’re worth the money. As I said, stick around the Alster and you should be fine.
We also spent some time in an Airbnb in the Winterhude neighborhood, just south of the Stadtpark. Again, just around the Alster, particularly on the east side, should be good.
In terms of sights, it’s been awhile since I’ve been to the Kunsthalle and Decorative Arts museums, so I don’t have enough memories.
– You should definitely have an Alster tour and a harbor tour on your agenda. If I only had time for one, I’d pick the Alster tour, but both are worthwhile.
– The Philharmonie in the harbor is now open, and while it’s hard to say a $1 billion is worth it, everything I’ve heard from family members who’ve attended say that it’s a spectacular place, worthy of all the hype. (Tickets for concerts there, while hard to come by, are generally cheap.)
– Related, there are a number of musical theatres in the city, so again, if culture’s your thing, Hamburg is good for that.
– Nearby the Philharmonie is the Miniature Wunderland, which is an incredibly complex, multilevel train set. I’m not doing it justice with that description, you can look at the website. Might not be your thing (it’s not exactly mine, either, but was very impressed with the detail when we visited), but if it might be your thing, it’s worth exploring.
– The city hall (Rathaus) offers tours in English, but you can also wander through for free if time is limited or you don’t appreciate fine government buildings. 
– There’s a museum about the city’s history, which was good (though it’s been awhile since I’ve seen it).
– Relative to city history, there are several famous churches which have been repaired to varying degrees since the bombing in WWII which are also sobering to visit.
– The Stadtpark is very nice; it’s been too long since I’ve seen Planten und Blomen. And a walk around the Alster (at least part of it) and up and down the Jungfernsteig / Monckebergstrasse is very important.
I understand why Hamburg sort of gets passed over by American tourists, but it is sort of weird that Germany’s Chicago is basically unknown on this side of the pond. But I think it’s a great city, wonderful for walking, with an incredibly friendly U-bahn/S-bahn system for moving around the city. You should be able to find 2 full days’ worth of stuff to do. Hope that helps.