Just one thought to start off. If you are a beginner, bad slopes can be a disadvantage ...
I would have a look at Germany, and more particularly the Allgäu region. You have some smaller resorts over there, which are a bit cheaper than resorts in Austria.
In the French Alps, places like Chamrousse and Saint-François Longchamp are not too expensive. The last one is a very nice place for beginners. Look for a flat and do some self-catering to beat down the price.
Northern Italy, with the exception of South Tyrol (Alto Adige), should not be too expensive either. Here I am thinking about Lombardia but also the Trentino.
If you want something really cheap, I would not go to the Alps. Czech Republic or the Tatra mountains could be a good choice.