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.

Hostels usually charge around 15 euros per person, depending where you are staying. In order to book a hostel you may need to have a Hostelling International card, which you can get at the hostel while in Europe. There are other hostels that don't belong to the HI organization and are just as good.
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.