Finding accommodation is usually a balance between quality and cost. A quick online search will give you some ideas of options in the area that you'd like to stay. With more and more online reviews available, it is a good idea to read what other people have to say about hotels, B&Bs etc. There are lots of different review websites available, a good example is TripAdvisor. When you read other people's reviews remember that they may have different standards to you. It is also a good idea to check the date of the review as the quality/cost of the hotel may have changed since the review was written.
You could also look for accommodation via the local 'Tourist Information Office' near to the place that you are visiting. If you are visiting a university town or city, the local university may be able to offer you cheap accommodation - particularly out of term-time.
Other accommodation
Airbnb (Range of different types of accommodation, often in someone's house)
Camping and Caravaning
Again, the best place to start is doing a quick online search and try to find some reviews if possible.
Independant Top 50 Campsites - this article is from 2006, but it is great for ideas. Even if you want to stay elsewhere, the article gives you an idea of the facilities that that different campsites may have to offer.
If you haven't been camping before, read this ' Beginners Guide to Camping'. If you need to buy a tent, sleeping bag and other camping equipment, you could try Millets on the High Street, Bangor, or you could visit the village of Betws-Y-Coed where you will find several outdoor shops.