Hamburg

Try these sites for ideas. Hamburg has a lot to offer. If you search this site, there are several posts with great ideas in the past couple years. I personally think the RS Guide misses the mark with his suggestions and rankings.
https://www.tripadvisor.com/Attractions-g187331-Activities-Hamburg.html
https://www.hamburg-travel.com

Bremen or Lübeck are good day trips (or overnights).

What are your interests?

What’s he best website to buy train tickets or train passes?

In Germany, alwayus use DB site or download the app https://int.bahn.de/en/

This is the site for the metro https://www.hvv.de/en

I use the DeutscheBahn site.
https://int.bahn.de/en

You can check if your travels make a regional pass worthwhile:
https://int.bahn.de/en/offers/regional

For a day trip a visit to Lübeck is a must.

In Hamburg, here are a couple depending on your interests:
Miniatur Wunderland
St. Nikolai bombing museum in basement
Harbor tour.

Even if you are not really into model railways, Miniatur Wunderland is an absolute must.
They also have a virtual reality experience which is quite something if you haven't done it before.
https://www.miniatur-wunderland.com/

There is also an illusion museum which is fun for an hour or so.
https://hamburg.museumderillusionen.de/en/

Open top bus tour.

Harbour cruise

Maritime museum.

We went to the opera (not everyone's taste)

Don't bother with the zoo - it is distressingly past its best.

We just bought metro passes from a machine at the station.

train rides of approximately an hour to smaller cities worth visiting

• Lübeck
• Stade
• Celle (by ICE)

Have a look at the AT A GLANCE section of the Hamburg page on this site:
https://www.ricksteves.com/europe/germany/hamburg

How far of a radius are you willing to go from Hamburg?

If somewhat over one hour, ie, 1.5 hrs, I would suggest going in another direction, go east, easily done logistically from Hamburg Hbf a day trip to Schwerin. Depart no late than 10AM, a direct shot . The famous and marvelous chateau / Schloss is well worth your time.

Lüneburg was one of the most beautiful towns we have ever visited in many trips. It is a short train ride away and truly a great choice for an easy day trip.

Daytrip. Ship starts in Hamburg. Island Helgoland.