If you want to stay somewhere a little more culturally authentic in Japan than an expensive tourist hotel, you will find a great option is a ryokan. There are many in Kanazawa, and the best way to find them is to ask at the tourist contact point at the central bus/train station, which is probably how you're getting to Kanazawa anyway.
They speak very good English there (I tried to speak Japanese but they hardly let me), and will both make suggestions and contact the places for you to find if there are vacancies. Just ask them for something quite small, and obviously with its own bath (ofuro).
You will be told to bathe in the evenings, and provided with yukata, the light summer version of a kimono that is used kind of like a dressing gown. You wear slippers when walking around inside the ryokan, leaving your shoes either downstairs or carrying them up to your room. However, if ever in a traditional house like this in Japan, NEVER walk into a tatami room with shoes OR slippers on. It is strictly socks only, and people will be quite shocked if you make this (unfortunately common) faux pas.
The bath is shared with other guests, but you shower beforehand and are therefore clean when you enter the bathtub. The water is very hot, but you get used to it and will probably find yourself red up to your neck when you get out. Very relaxing, and essential in the winter as the rooms only have a portable heater that you will hopefully figure out how to operate.
If you are unfamiliar with the Japanese traditional house, you should know that you sleep on a futon rather than a bed. It involves a flat mattress and a thick heavy version of a blanket, plus several other layers. Just get in between wherever you please. They are very comfortable, and I think good for your back too.
You are served breakfast in a common dining area, usually a tatami room where you sit on a cushion on the floor, and are served a traditional breakfast, undoubtedly involving rice, fish, and miso soup -- all food groups that cannot be avoided in this country, though I'm certainly not complaining.
Often there are also laundry facilities and the staff will help you figure out where to go in town. Ask for the best sushi places or whatever you're into and they'll probably draw you an elaborate map (though not necessarily a very useful one!)
Prices vary depending on the size and quality of the place, ranging from budget to luxury. I've stayed in some places where the heating hardly works and there's no room to put your luggage, and some where you get your personal assistant, green tea served at regular intervals during the day and a seven course meal after your evening dip in the hotel's private hot springs.