Which Alps should we visit given transportation limitations

A car would not be budget-friendly if you needed to pick it up [critical omission here--I meant to say "pick it up in Italy"] and drop it off in Germany, but I don't know how your itinerary plays out after Mannheim; I'm guessing you do not plan to return to Italy after Mannheim.

The core of your question--the mountain area to choose--requires an in-depth knowledge of rail lines, which I do not have. I see that you have at least two choices for a reasonably efficient route between Venice and Mannheim taking between 9 and 11 hours without detours; one goes through Como, Lugano and Luzern; the other, through Stresa, Domodossola and Bern. A key question is how much time each route spends burrowing under mountains so you see nothing at all for a chunk of the trip.

You could also diverge from the most direct routes in order to include the Benina Express, which would increase your travel time substantially. I will offer this comment from personal experience: The Centovalli rail line between Domodossola and Locarno is a very interesting ride; however, it would also add time to the trip.

Thank you!

From Venice fly nonstop to Munich and rent a car at the airport and drive to the Bavarian Mountains.

Tricky question to answer as there are several options and it all depends on what you consider the best choice.

Somewhere in Switzerland can be a good idea, an easy stop on the way. But, it is not know as a low cost destination. If that is important, I'd suggest Innsbruck, a good base for exploring the Austrian alps.

What Mary Pat said. Fly to Munich, rent car at airport, spend time in southern Germany or Austria, drop car in Mannheim. Enjoy your trip!

The main north-south train line between Venice and Germany goes right past the Dolomites and then through the Austrian Alps.

For the Dolomites, you could take the train from Venice to Bolzano ( one change at Verona), a journey of 3h30 minutes. At Bolzano you take a bus into one of the villages in Val Gardena, such as Ortesei. The beautiful Dolomite peaks rise up on either side of the village, with views from many hotels and lifts to take you right up to the trails.

Return to Bolzano by bus and take the direct train to Munich, a 3-hour ride. You can rent a car there to continue your journey through Germany.

For the Austrian Alps, take the same train from Venice/Verona, but stay on til Innsbruck. The mountains are all around, and it is a short train or bus ride to alpine villages.

If you prefer the Swiss Alps, there are a couple of train routes that go over the Alps instead of under them in a tunnel. Acraven mentioned the Bernina Express, but it is well out of the way for your trip. Another option is the Treno Gottardo, which connects Locarno with Luzern, avoiding the deep Gotthard Basistunnel. For this you would take the fast train from Venice to Milan, changing there for the short ride on a regional train to LocRno, where the Treno Gottardo originates. Or you could ride from Milan to Bellinzona and pick up the Treno Gottardo there, but you would have to be very careful to get the right train ( not the one through the tunnel). My preference would be to spend the night in Locarno in a hotel right on the lake, and continue to Luzern the following day.

Once you reach Luzern you can stay there and tour nearby mountains by train and boat, or take the train to the Berner Oberland area and choose a small village in the heart of the Alps.

But the Dolomites would be the simplest, and they are so beautiful.

One more option for your comparison:
Bad Gastein (at 1000 meters altitude above sea level) is on the train line Venice via Villach to Salzburg.
After passing the main ridge of eastern Alps through a 12 kilometer long tunnel your reach Gastein valley with a great variety of walking options and other outdoor activities.
Venice to Bad Gastein is about 4:30 hours per train (1 change in Villach).
Bad Gastein to Mannheim is about 6:30 hours per train (one direct train 1 per day) via Salzburg and Munich. Thus, it depends on your plans where to rent a car.

After visiting Venice, take the train up through the Brenner Pass to Innsbruck. The city is in the middle of The Tirolean Alps and the scenery around there is so easy to visit--and easy to get around.

I used to live in Innsbruck, and there were many, many peaks over 10,000' out my apartment window. We love to rent a car and get up on the mountainside going from village to village.

It's easy to catch a train up to Munich and on to other German cities along the Rhine. It's also an easy ride to the ever popular Salzburg too.

From Venice, the dolomites and then on to germany seems to be the easiest option.