In reply to C Witter:
I love your idea of a "straightforward approach" which is entirely not my idea of what that expression would mean!
I have a German friend who is great at finding random places to climb and I've been to loads of places with him in the Alps. But. I can't remember the names and my guidebook pile is 200 miles away, sorry! The thing is, there is just masses and masses of climbing in the Swiss, Italian and French alps that is not well-known but you are probably not going to find out about it on UKC because (without meaning to offend), the answers given so far are all very well known areas that are (in my experience) very busy with the exception of Salbit which I've not heard of and Courmayeur which I have not spent enough time in to know.
If you ask on UKC for winter sport climbing, you are going to get Siurana, Leonidio, Kalymnos, San Vito Lo Capo, etc. but there are countless nice sport places that are just not known in the UK and where you might have a much quieter and off-the-beaten-path experience, but it will be hard to elicit the names on UKC.
As you said you could speak Italian, I'd suggest asking the same question on whatever the Italian version of UKC is.
The amount of granite and limestone slab climbing in Switzerland is eyewatering but on a budget you want to make your plans very carefully. Basically buy all your food before you enter Switzerland and take your car and tent for some wild camping (the rules and enforcement thereof relating to wild camping in Switzerland vary enormously from area to area though, so be sure to check on that first.