"is there only 1 main train"
"is there basically 1 train we would take from Hamburg to Dresden"
did you mean train or train station?
As Chris said, Hamburg has the large Hauptbahnhof and the slightly smaller Altona (actually almost as busy but many of the suburban trains are out of sight underground). It also has Dammtor in between. The mainline is an inverted U with Dammtor on the bend at the north, Altona at a dead end at the west and Hbf a through station at the east, the river runs west east about half a mile south of Altona and Hbf. Most of what you'd want to see is within this curve and the river, it is walkable but if raining the underground railway has frequent stations - and some of it is actually elevated with views of the river!
You need to be careful where you stay because the rivers and old harbour basins break the city up and what a mapping system says is close by could be a long walk to get around the water. I stayed in a nice place south of the river in Wilhelmsburg but I wouldn't recommend it because you couldn't walk into the centre, to the tourist area it was 2 miles straight line and about 4.5 on foot! When I stayed in March I went to the Ibis Budget in St Pauli which is almost new and pretty much all you need, it's about the same distance from all 3 mainline stations, ie slightly too far to carry luggage! But close to St Pauli U-Bahn station. I got a good deal by booking several weeks early via https://www.hrs.com/
Fast trains from Berlin run at least hourly, most call at Berlin Hauptbahnhof (low level) and Berlin Südkreuz. All call at Hamburg Hauptbahnhof, some at Dammtor and Altona. Journey is about 2 hours.
Hamburg to Dresden usually requires a change in Berlin but there are some direct trains, notably 08:51 (only from Hbf) and 12:51 from Hbf and slightly earlier from Altona, these take 4 hours to Dresden.
All fares are much cheaper booked in advance (penalties for later changes). https://www.bahn.com/en if you book here you can take the ticket as a PDF and print at home. Payment by the usual cards or Paypal!