Train travel from Varenna to Salzburg Austria

Strangely phrased question. There are not multiple rail systems, in the sense of more than one route from A to B but owned by different companies. What there are is different companies in different areas, usually national companies, but they do not overlap.
International services are usually jointly operated, but may be operated by a jointly owned company, or even by a competing operator. But these will all be using the same tracks so it is the same "rail system". And no matter how many companies are involved, you can usually just buy one ticket from A to B.

The real question is what trains go from Varenna to Salzburg. You can look this up on the DB (German Railways) site: https://www.bahn.com/en/view/index.shtml
This lists most trains in Europe, but only sells tickets for trains within, to or from Germany.
For buying tickets, go to the actual operator, either Trenitalia (Italian Railways): http://www.trenitalia.com/tcom-en or ÖBB (Austrian Railways): http://www.oebb.at/en/

I looked it up on the DB site, and it gave me lots of options, for example:

Varenna-Esino dep 10:37, Milano Centrale arr 11:40
Milano Centrale dep 12:05, Verona Porta Nuova arr 13:28
Verona Porta Nuova dep 13:50, Brennero/Brenner arr 16:52
Brennero/Brenner dep 17:33, Innsbruck Hbf arr 18:13
Innsbruck Hbf dep 18:17, Salzburg Hbf arr 20:03

Should you wish to break up the long train trip and spend the night somewhere, both Bolzano (larger) and Bressanone (smaller) have very nice historic districts. Bolzano's is considerably closer to the train station. The Iceman is housed in a museum in Bolzano.

Note that this area is sometimes referred to as "the Dolomites", but the rail line runs through a valley, and it can get very hot in summer while it's lovely up in the mountains. I wouldn't book a hotel without air conditioning for June unless making the arrangements late enough that I had access to a reliable weather forecast. Unfortunately, many otherwise lovely hotels in Bolzano and Bressanone are not air conditioned. Nor are many (most?) of the restaurants.

I would take the opportunity to ride the very scenic Bernina Express route through Switzerland---not necessarily the named scenic train, but a regional train. The route would be north from Varenna to Tirano on the Swiss border. Then north on the Swiss train toward Chur, and Sargans,nowhere you catch a train to Salzburg.

This takes a bit longer than the route via Milan and Verona, but the scenery is worth it if you are interested. Any route is long enough that I suggest you break it up into two days. However, if you must do it in one day, the shorter route via Milan and Verona takes 9.5 hours and would be a better choice.

Thank you very much for the information. I will be traveling a similar path in May 2017. The German train website was very helpful. It looks like we will still need to make seat reservations. I remember standing in line for a long time years ago to make those reservations, I was hoping there was an easier way to do it nowadays. But I'm not seeing it.

Are you using a Eurail pass? The only train needing a seat reservation is the Milan-Verona leg. Buy it ahead of time at Rail Europe. If not using a pass, simply buy the ticket and it will automatically come with a seat reservation.