Italy-Germany-France?

Your question is really too broad for us to answer. What sort of places would you like to see? Mountains? Coast? Lakes? Rivers? Large cities? Small towns? Historical sights? Art museums? Churches? Are you into wine? beer? active sports?

Trains will almost certainly work fine, with occasional use of boats and/or buses.

You can access most places by rail, but once there, a car can sometimes be nice if you want to get out into the country (Normandy, the Loire Valley, or the Dordogne as examples in France). Like acravan said, my first response was, what do you want to see. Obviously Switzerland lies between these and has many wonderful places to stop. But yeah, cities, villages, lakes? Have you looked at a map just to get a sense of the train route and what would be on it? Have you read anything in guide books for the areas. Its hard to go to Europe and make bad choices ;p Its a matter of how much time you have and what your priorities are...

It would be useful to know if this is your first visit to Europe, and whether you imagine having another opportunity in your lives to visit. Besides the previous replies, it's not possible to advise without knowing approximately how much time you might allow for the trip. For example, it might be better to start in Paris or Brussels, or even Hamburg (just to name some air hubs) - depending, of course, on your previous travel and interests. You also need to fill in your home state and country, so we have an idea of how annoying your jet lag and connection options might be. Do you know what an "open-jaw" flight itinerary is?

I agree that it is not possible for an internet board to provide what you really need for such an open ended question. Our host's books are in a lot of libraries, and he certainly covers the highlights of basic European travel. You will have a much better time if you can incorporate your own interests and backgrounds into your itinerary. You also need to make a general decision about how often to change hotels and how many attractions you want to see per day. There is no need to rent a car.

Is traveling by rail an efficient way to do that trip? I'd rather not HAVE to rent a car.

Travelling by rail is efficient for big cities and most towns. Assume you can get any town to any town by train unless told otherwise. In cities car is a useless lump of metal which you have to pay to park. And renting a car in one country and returning it in another has a large surcharge.
Start by reading this webpage, followed by the rest of the website: https://www.seat61.com/Europe-train-travel.htm

If you want small villages and off the beaten track, in most places a car is better. Exceptions include Switzerland.

But (stating the obvious), Europe is a big continent. As the others say, you need to give us more hints as to what (and where) you want to see. I suggest you read some guide books. They contain hundreds more pages of guidance and suggestions than we can type!

Knowing how long you are there is a key bit of info, if you want salient advice from the forum participants.

The advice will be quite different for a 10 day trip vs a 30 day trip.