Hotels

No you don't need (or in my opinion, want) to stay near the train station. The trams run throughout the city and are different than the train.

Did you try searching the forum for hotel recommendations in Amsterdam? There are dozens of posts on this subject.

Amsterdam also has a Metro and buses.

Sights are rather spread out, and as in Venice, you may have to diverge from the shortest, as-the-crow-flies path in order to find a bridge across the next canal. The need to monitor bicycle traffic is also likely to slow you down. A location relatively convenient to your sightseeing targets can be very helpful.

It would be smart to mark the sights you want to see on a map (Google or otherwise) to get a feel for the city's layout. That will help in evaluating the practicality of the lodgings you are considering. I'm on Day 11 in Amsterdam at the moment, and I have walked around a lot. I haven't seen a bad neighborhood, though some areas have nicer or less nice architecture, more or fewer food options, etc. The area immediately south of the train station wouldn't be my choice; it always seems crowded, and it has more than its share of junky tourist shops.

Are there hotels that are near public transportation to move around Amsterdam. Or should we just look at hotels near the Amsterdam Central Train station? Or is the trams a separate type of transportation?

Amsterdam is criss-crossed with public transportation options. Much of it connects into Centraal station, but it is very easy to use simply the tram system or add in bus and Metro options to the mix as well. All are part of the same network and use the very simple tap-on/tap-off payment system.

The tram system seems the "easiest" with tons of stops, but we've taken the bus from the airport to our hotel (as well as the train a different time), and I like to add the FERRIES that run from behind Centraal as a FREE and FUN option.

So, YES - pretty much all of the touristed areas & hotels in Amsterdam are on or very near a public transportation option - usually MULTIPLE options.