You can't ski between any of them but there are shuttles available. Buttermilk and Snowmass would be the best of the four mountains for beginners.
Aspen Mountain is in the town of Aspen, and Snowmass is about a 20 minute drive and has a small town of its own. Buttermilk is on the road between Aspen and Snowmass but there isn't much there besides the mountain.

Aspen mountain is a pretty expert terrain. Buttermilk and Snowmass will be great for learning--particularly Buttermilk.

Unless I missed it, I don't consider what Snowmass has "a town of its own", small or otherwise. It's a ski area with all the related glove-and-hat stores and condos that come with a ski area.

The shuttle between the mountains is free (or was last time I was there).

Buttermilk is indeed a great hill for learning. Snowboarding in particular can be pretty hard on your body the first time out, so the gentle slopes there will be appreciated.

There are four ski areas associated under the Aspen banner. In Aspen, itself, is Ajax Mountain commonly called Aspen Mountain which is geared more for the expert skier. The Highlands which is on the west side of Aspen focuses on the expert and strong intermediate skier. Buttermilk area. a few miles north, traditionally has been the learning or beginner mountain although they are tried to break that reputation. In twenty plus years of skiing at Aspen I have never skied Buttermilk. Further north along the road is Snowmass. A massive ski area that is roughly 15% beginner, 10% expert, and 75% intermediate which ranges from big, wide cruising slopes to steeper, some trees and more demanding intermediate. Snowmass is my favorite ski area is all of Colorado. The past 10+ years have seen a major redevelopment of Snowmass around the Elk Mountain area. May not want to call it a town but a number of good restaurants and entertainment in the area. If you cannot find happiness at Aspen you cannot find it anywhere.

So it looks like the Snowmass Village is still under development? Does that mean lots of construction equipment/noise?

Not particularly. First, very little exterior construction goes on in the winter -- too cold and too snowy. The most would be an inactive construction site. Second, the redevelopment has been slow - mostly one building at a time. The bulk of the work has been completely but since I missed last year I don't know what is currently under construction. But will be there the first week of March. Probably not much different than Aspen as there has been a number of new construction projects recent in Aspen. It is rare for a ski area not to have some on going construction.