"Wool" is a very broad category that pretty much means "material made out of some kind of animal hair". And since there's quite a few animals with hair, there's quite a few types of wool. I suggest trying high quality cashmere (note: cheap cashmere is typically mixed with other types of wool).
It depends on the wool. Finer wool like Merino wool tends to be nicer to wear directly on your skin. But you don't have to do that. You can and should wear cotton underwear. The extra layer keeps you even warmer and means you won't have to wash your outer wear all the time (wool should not be washed more than is absolutely necessary and ideally not at all). You'll also find heavier garments encase the wool in an outer material like cotton. You can get cotton duvets like this and they are wonderful. You can use them all year round. I will never go back to polyester again.
Don't you find wool not stretching is a real problem? I struggle to get well-fitting wool socks that I can also get around my ankle, as they have no stretch.
Did you know people used to use sock suspenders to keep their saggy wool socks up before we started to use synthetic fibres for socks?
It's not a problem if you find appropriate quality and thickness. One example is Icebreaker Ultralight socks. I use them in summer and they are rather thin.
I've never tried bombas wool, but I have a pair of bombas socks which were gifted to me, and other than being quite tight, they are serviceable. Make sure you get a large enough size.
Bombas are one of the few brands I would recommend to anyone because they donate one pair of socks for each that you buy.
Always use cotton as a first layer, because you can just put it in the laundry. Cleaning my woollen coat is more difficult than changing a cotton shirt underneath it.
Also: most “wool” is polyester because it’s like 10 times cheaper.
Cotton remains damp after you sweat and eventually also stinks of sweat. It's not a good base layer in cold weather. A lot of merino wool products are easy to clean and maintain. I have never seen polyester called wool.
This is good - potentially even life saving - advice if we're talking about doing serious activities outdoors. And in that case, even wool is probably best replaced with synthetics that are lighter and dry quicker when wet.
For around town, though, a cotton base layer and wool works pretty well.
I must side with the other replies to your comment.
I have been wearing cotton as my top base layer for a while, and it is not great at staying dry or drying out while damp.
I'd certainly choose it over any synthetic fabric, but once you go over that low mark, just about anything is better, e.g. wool, linen, help, silk, etc.