Although hotels can usually even be found in smaller cities they are quite expensive even more so in bigger cities cheaper possibilities in big cities are youth hostels and in smaller towns you can often find families renting flats in bed and breakfast style look for Pension or Zimmer Frei signs for €15-25. In the countryside many farmers will rent out rooms for a couple of nights, both officially and unofficially. To find a place to stay, simply knock on the door of a farmhouse and ask - if they don't have a room they'll probably know someone nearby who does.
You can also find a lot of camping grounds some of them are open the whole year round but while they are exceptionally clean and often provide additional services, they are also a bit more expensive than in other countries in Central Europe.
Austrian law requires anyone to register at their resident address, even if it's only for one night and even if it's a campsite.
Hotels will therefore ask you to hand over your passport or driving license and may refuse to give you accommodation if you don't have any ID on you. Don't worry too much about handing over your passport. In many countries, such a practice would raise concerns, but in Austria, it's a standard procedure. Your passport will be returned. If you stay in private accommodation for longer than about two weeks, you should obtain a document of registration Meldezettel from the local registration authority Bezirksamt or Meldeamt, usually located in the town hall. This document needs to be signed by the owner or tenant of your accommodation. Failure to present this document upon departure could cause difficulties if you have stayed in the country for more than two or three months.