Italian Lake district by train?

You can get to many of the lakes. There are many lakes, and all have different geography. Where (which lake and town) do you actually want to go to.
Sometimes rail lines go up one side and not the other, or vice-versa, or just connect to the end of a lake. But the good news is in many cases lake boats connect with the train services.
You need to decide which lake or lakes you intend to visit, and look at a map which shows rail lines and stations (i.e. not Google maps, which is totally road-orientated). Then look up sample times and prices on www.trenitalia.com from Venice to wherever, and between locations on the lake(s).

Edit: I did a quick search and found 2 rail maps of Northern Italy.
Here: http://www.bueker.net/trainspotting/maps/italy/italy.gif
and here: http://www.bueker.net/trainspotting/maps/milano-area/milano-area.gif
They are slightly out of date and don't show stations, but at least they show all the lines and should give you some idea of what is possible. You can then check individual towns and villages to see if they have stations.

Edit 2: I just found this. A zoomable map which shows trains and bus routes: http://www.öpnvkarte.de/

http://www.rome2rio.com/

This will tell you want you want to know, I think. It's served me well in figuring out logistics.

For example, I was doing some research today and found out that from the Verona train station--where we're planning to base for a few days on our next trip--it's only a 20 minute ride to Desenzano on the south side of Lake Garda, and then from there a ferry can take you to various stops around the lake, where there are some ruins from the Middle Ages and before that.

Seems to be a painless and perfectly pleasurable way to travel...