Parts of Europe, such as Ireland and the Netherlands, are extremely suited for bicycle riding. Many hostels in these areas rent bicycles, or can tell you where to find someone that does. Bicycling is a nice, relaxing way to really see and area while still covering ground. In the Netherlands bicycles are everywhere, both in the countryside and big cities like Amsterdam. Amsterdam has bicycle lanes on the streets, with special walk-don't walk signs for the crossings.
Trains are the basic way that most travelers get around Europe.
Trains connect just about every single town in most areas. Train travel is extremely convenient and well-designed. Look out for tourist information booths located in city train stations. Trains are usually filled with local people as well, and can be a great place to meet interesting locals. Do not be surprised to see pets on many trains.