That's good advice, stumps. Often the smaller places are so much better to visit, or use as a base.
The first thing to decide is how to get around, which can depend on how many people you're travelling with. If you're a couple, I'd go by train with a Eurail pass. Get on and off easily, usually in the middle of town. It's a great way to get around Europe. I'd always drive in America, but train in Europe.
As for what to see and where to go, it depends on your tastes.
Some less tourist-frequented parts of Europe I'd recommend in each country are"
Spain - go to the northern part (Basque country), San Sebastian - best tapas in Spain, Peaks of Europe National Park - snow mountains, alpine meadows, hiking, fishing villages, beaches, caves, old towns.
Spain- the Costa Blanca, small villages like Altea, great scenery, sun, beaches, dinner.
France - Strasbourg (french baguette with German sausage!), Brittany, Carcassonne - one of the gems of Europe. Go to Annecy, a mini-Venice on a lake in the Alps. Chamonix and the cable car to Mont Blanc.
Switzerland - Lake Thun, St Moritz - wonderful, Lucerne, even though it's busy.
Austria - Don't miss Hallstatt, a world heritage village and superb. Salzburg, like Lucerne, busy but don't miss.
Italy - anywhere. The lakes in the north - Como, etc. For a great day, take a boat trip from Lugano (?), Locarno (?) of Lake Maggiore, visiting the islands in the lake.
Eastern - Prague is good, but Budapest is well worth visiting. If you can get as far as Istanbul, it would be one of the highlights of Europe (and you can go to Gallipoli).
Germany - The Black Forest area. Hitler's retreat, Berchtesgarten is scenically superb. The Romantic Road, including Rothenberg-ob-der-Tauber, is sensational. The Moselle valley, as Pseudo said, is wonderful, including Trier.
Holland - The Alkmaar Cheese Markets are great to see.
Scandinavia - Copenhagen is well woth a visit, Oslo the same. Stockholm is a long way, but the Swedish south coast is interesting. Taking the local buses and ferries from Stavanger to Voss and the Fjiords in Norway is unforgettable.
UK - The Lake District, Clovelly, Cornwall, Wells in Somerset, Cambridge, Canterbury, the mountains and valleys of Wales, the Scottish Highlands, the fishing villages of the Scottish North-West, anywhere in Ireland, Clear Island, Arran Islands and don't leave out Northern Ireland, it's wonderful.
Yes, I like Europe
