From Basel, I recommend learning in Feldberg, Germany. It is a one hour drive so you can come/go same day with no overnight expense. Free parking, day ticket for the lifts are E. 22-29 depending on the area, with pay as you go options for those early days when you only take the shorter lifts very few times in one day that are cheaper. Ski schools on site have 2 daily lessons starting at around 10 am and 2 pm. Pack your own lunch for maximum savings.
Within Feldberg, once everyone can at least get down the bunny hill in one piece, I recommend the Belchenbahn lift (still in Feldberg, but a different hill), because it has gondola lifts and a very gentle sledding slope of about 5 kms that is great for just learning to be comfortable on the skis without ever speeding up too much. It's like a giant bunny hill. This area is also much calmer, less people, although the ski rental and ski school are about a km down the road from the actual ski area.
You can do this a few weekends in a row to get the hang of it before booking the full ski experience at a higher altitude resort.... although you can very well spend a week here.
You mean like the new pair of boots I hired then bought there (some 10 years ago) or the new pairs of skis that I hired for my kids and were brand-spanking new... (last season).
Don't assume that all hire stuff is old. If you go early enough, they'll be hiring out new kit. If you mention you are looking to buy long-term and explain that you want to try first, they might even pull out a new pair for you to hire...
Just to back up what Carlos has said, I know a number of people who've rented there from early in the season and had brand new kit, and quite a few that have then ended up wanting to keep it and buying it an the end of the season, for an overall cost still lower that they'd have been able to buy it for in the first place.
Seems to make no economic sense, but in fact the ski market is such that shops pay very little for the skis if they're going to use them for rental. Special agreements with the suppliers that they're not simply going to sell them new below retail price, but once they've been used for a season they can do what they like with them. So your 165chf for a basic set that probably costs the shop about that will already pay their costs, then you pay perhaps another 200chf at the end of the season which is all profit.
(Note those figures are made up, so don't quote me).
If you rent equipment it will be cheaper short term than buying, but if you are likely to go several times then it might be worth buying at least the adults' equipment (not sure about the kids' though, because they'll grow between each season). It is probably worth renting the first couple of holidays you do, so you find out whether the kids like it enough.
Austria is definitely cheaper than Switzerland.
Go to a smaller resort rather than one with hundreds of kilometres of pistes, they tend to be cheaper.
See if it makes sense to buy half day lift passes rather than full day/week ones. The snow often goes off in the afternoon anyway, so I prefer to ski in the morning. You can do other stuff in the afternoon.