These are my personal favorite ski destinations, with nice resorts and reasonably easy slopes in Europe:
A.If you want to ski on the Mont Blanc: Chamonix (FR) or Courmayeur (IT). Nice resorts, reasonably updated lifts. Nice multicultural/younger crowds. Excellent food at reasonable prices. The offer of apartments, AirBNB and Hotel rooms is very large, so even in peak season you can find something for almost any budget. Downsides: cars everywhere during peak season! It’s a mess to move aroundunless you walk (but the villages are pretty large so everybody uses the car).
B.If you want to ski on the Matterhorn / Mount Cervin / Cervino: Zermatt (CH) or Cervinia (IT). The slopes are fantastic and the domineering view of the Matterhorn is absolutely intoxicating (of the two sides, the Swiss one is probably more beautiful). The slopes are well groomed and the lifts have mostly been renovated in the last 10 years. Ski teachers are multilingual and very kind (but cost like a London finance lawyer!). Zermatt is completely car-free and you can reach it very easily by train from Zurich or Milan: fantastic. Downsides: Peak season is after January 6th (Orthodox Xmas) because of the inflow of wealthy Russian visitors (in January prices are very high). Also: there is basically no nightlife (at night the silence and gentle light system throughout the city will make it look like a fairytale Alpine village) and the people skiing there are mostly the age of your parents.
C.If you want to ski on the Dolomites: Cortina d’Ampezzo (IT). Excellent slopes with very nice views over the Dolomites. Food is good and the village is beautiful and well-maintained. Hotels are a bit old but with charme. The crowd is mostly Italian, with the occasional foreigner. Downsides: Cortina is very expensive (not just for Italian standards: I compared the prices in January and they are generally on par or more expensive even than Zermatt)and it can only be reached by car or coach bus. Also, you will most likely need a car to move around as the city is pretty wide and some of the best slopes (San Cassiano, Alta Badia, etc.) are only reachable by car. The closest slopes are Faloria (walking distance from the city center) and Socrepes (take the n. 3 bus). January/February is peak season, so prices are normally driven up too.
Last December I met in Zermatt (by many regarded as one of the best ski resorts on the planet) a 19 years old boy from Tel Aviv.
He was very well educated and friendly and we started chatting on the lift: after a bit he explained that he was admittedly money-less as he was studying Mathematics at University and was getting ready to serve in the Military in the coming year.
He added that he loved skiing and normally he was able to ski only in Bulgaria as that country had the cheapest ski resorts reachable from Israel.
I looked at him a bit surprised, given that we were chatting in one of the most expensive ski resorts in Switzerland.
He then explained that this December he planned in advance and with a friend he was able to book a Ryanair ticket from Tel Aviv to Bergamo (close to Milan) for 10 Euro only.
Once in Bergamo, he booked a seat on a very cheap bus to Aosta for just a couple of Euros. In Aosta (north west of Italy), he booked a cheap hostel there and on a daily basis moved to Cervinia by bus where he was able to ski for less than 50 Euro a day.
In low season (before Christmas) you can make the most of your ski days as the slopes in Europe are basically empty.
From Cervinia (Italy) you can ski and cross the border with Switzerland (crossing the Matterhorn Glacier, right in front of the majestic Matterhorn) and reach even Zermatt, if you want to do so.
This is just an example that even if skiing requires dedicated equipment, travelling, ski-passes, etc., if you are passionate for this sport, with a bit of planning you can do it also on a budget.