Recommendations for train travel and walking

You can set up something is nearly all European countries. The except might be Spain since the rail system is not as extensive as most of Europe but Spain has an excellent bus system. You could do something similar in France, Italy or even Germany. Just a matter of where you want to be. In Germany you have a lot of great rivers.

aberry, you sound like my twin - if I had one.

Best place I know of lies south of Koblenz on the Rhine River. Towns every few miles. Railways and stations in nearly every little hamlet on each river bank. Scenic trails on the cliffsides above each river bank, with a constantly-changing landscape (open fields, forested slopes, rocky cliffsides, vineyards.) River cruise boats running upstream and downstream. And ferries that connect the river banks here and there:

http://www.loreley-info.com/eng/rhein-rhine/walking-hiking.php

You can fly into Frankfurt and and after a one-hour train ride arrive in any of several great little riverfront base towns. Boppard and St. Goar (both with ferry crossings) have waterfront accommodations and are my personal favorites.

We have done this, for a week not several, in France (Chamonix), Germany (Garmisch-Partenkirchen, and Italy (Dolomites, various villages). The Dolomites are not well-served by train, but the local bus system is excellent.

I am thinking of Mayrhofen in the Zillertal (Austria) for next time.

In Italy you can fly nonstop from New York to Venice and see the surrounding islands plus there are good train connections to Padua (30m), Vicenza (1h) and Verona (1h 30m) for the day.
You can also fly in and out of Milan which is probably ITs cheaper airport to fly in and out of and take day trips to Varenna (1h 15m), Bellagio (must take a 5m boat ride from Varenna), Verona (1h 15m), Vicenza (1h 45), and Florence (2h [make sure you reserve tickets for the Uffizi Gallery as soon as you buy your plane tickets]), Stresa (1h 30m) and Lugano, CH (1h 30m).
Another idea is to fly in and out of Milan and take a train to the central station and then a direct train to Monterosso (3h). You can take direct trains and/or boats to Vernazza, Corniglia, Manarola and Riomaggiore and trains to Pisa (1h 15m), Levanto (10m) and Sistri Levante (45m).
You can fly in and out of Rome to make that your base and take day trips to Orvieto (1h 15m), Civita di Bagnoregio (need to catch a bus from Orvieto), Florence (1h 45m) and Naples (1h 15m).
Another great place in IT is to stay in Positano and take boats and busses to get around the Amalfi Coast, Paestum and Capri. You’ll need to fly in and out of Rome.

Not to discourage anyone from visiting Civita di Bagnoragio, it's a very cool place to visit. From Orvieto it is about a 4 hour walk... Most folks will opt for the bus or a taxi from the train station in Orvieto.

As has been noted, train travel works extremely well in most of Europe. Rural busses and taxis connect the dots where the trains don't do. For planning such a trip I usually decide where to go and then figure the transportation out.
Having said that, a car is a blessing if you want to deeply explore regions. In Tuscany I rent a car in Orvieto and stay at an Agraturismo. In the Black Forest region I rent a car in Munich and stay in one of the nearby towns. To travel through the Burgundy or Champagne region or to visit Southern Belgium I get a car in Paris. A car opens up options.

As mentioned, you can probably do that in most European countries. So the first question is, what kind of scenery are you looking for? Mountains? Forests? Coastal areas? Would you like more populated areas or less populated areas? What kind of climate?

In DE you can fly in and out of Frankfurt and sleep there. One day you can take a 1h 15m direct train to Bacharach and catch a boat to St Goar and visit the castle. In St Goar get bak on the boat and go up to Koblenz to take a train back to Frankfurt (1h 30m).
Other places to go from Frankfurt are Cologne (1h 15m), Würzburg (1h 15m) and Baden-Baden (1h 30m).

Thank you Mack, I just edited my post.

If you go to Belgium, it would be more convenient to sleep in Brussels. I would also go the weekend closest to Aug 15 to see the flower carpet in the main square. From there you can take day trips to Ghent (30m), Brugge (1h) and Antwerpen (1h).
If you go to FR, you can fly nonstop between New York and Nice and take day trips to Villefranche-sur-Mer (10m), Monaco (30m) and Antibes (15m).
In the Netherlands you can sleep in Amsterdam and make day trips to Haarlem (15m), Delft (1h 15m) and Rotterdam (45m). There’s also Hoorn (45m), Enkhuizen (1h 15m), Alkmaar (45m), Leiden (45m), Arnhem (1h 15m) and Utrecht (30m).

Other day trips from Frankfurt include Heidelburg and Mainz.

hey hey aberry
putting this out there if it helps.
gpsmycity.com
self guided tours in european cities. click country , city, scroll down and show you all self tours, with places of interests. i'll let the others advise you of different area. enjoy
aloha

I may have misunderstood, but I got the impression from the original post that aberry is looking for a good area for a hiking holiday, a place from which trains could be used to and from the hiking locations, not so much a place from which a lot of day-trips to other cities and towns could be made conveniently by public transportation. I do a lot of hub-and-spoke day-trips by train and bus, but I don't call my trips "train and walking holidays".

@acraven: You're not the only one, I also interpreted it a as a hiking trip.

Thanks so much for the great ideas from everyone! I am loving all the great recommendations! Sorry if my original post didn't make it clear, but yes we are looking for recommendations of a good home base town to rent a place in, and from there we'd walk and take trains from village to village etc, which is what we did in Switzerland. Thanks again!