If you are on a budget then you will most likely be staying at hostels. Do not worry about the horror stories that some people have about hostels, I have never had a problem when staying at one. They are a great place to stay and you get to meet so many other young travelers who are glad to get to know and willing to exchange information about their travels.
The hostels all have clean sheets, so you wont have to bring any of your own. One thing that I do recommend is bringing shower sandals just because some of the showers are multi-person and you don't want to get any fungus. But seriously, hostels are clean, friendly places.
Sometimes, especially during the busy months of July and August, hostels will book up. If you really want to stay at a particular hostel you can book ahead of time on the internet or over the phone. Be aware that some hostels operate on a first-come-first-served basis and do not take reservations ahead of time, so you may have to stay one night in a hotel if you are unlucky.
Here is a list of hostels and accomodations that you can find in each city:
Austria - Salzburg,
Vienna
Belgium - Brugge,
Brussels
Croatia - Zagreb
Czech Republic - Cesky
Krumlov, Prague
Denmark - Copenhagen
England - Bath,
London, Oxford
Finland - Helsinki
France - Biarritz,
Lyon,
Nice, Normandy,
Paris,
Strasbourg
Germany - Berlin,
Leipzig, Munich
Greece - Athens
Hungary - Budapest
Ireland - Cork,
Dublin
Italy - Cinque
Terre, Florence,
Rome, Siena,
Venice
Netherlands - Amsterdam,
Rotterdam
Norway - Oslo
Poland - Gdansk,
Krakow, Warsaw
Portugal - Algarve,
Lisbon, Porto
Slovenia - Ljubljana
Spain -
Barcelona, Bilbao,
Granada, Madrid,
San Sebastian,
Sevilla, Valencia
Sweden - Gothenburg,
Malmo, Stockholm
Switzerland - Bern,
Interlaken,
Zurich