On this Sunday morning, I took a train from Glasgow to Lanark, a direct connection taking 55 minutes. At the rather desolate Lanark station, there was a bus waiting that took me to the New Lanark WHS within a few minutes. Measured by the huge car park, this is a very popular place for summer outings.
This site's greatest asset is its really lovely setting, near the river that supplied the necessary power for making the mills work. A good place to develop into Utopia! The buildings in this tiny village are well restored, and some of the houses even are in use as modern apartments. You can have a look inside several houses, the mills, the school, and a shop. All in all, it's a nice place to walk around for an hour or two.
On the minus side, I didn't find the exhibitions too great. There's a lot to see and do for children, but not so much for people over 12.
Like some of the Derwent Valley Mills (another UK WHS, related to this one and very much alike in ambiance) some of the buildings are used for uninteresting shops. And why this strange award gallery that you have to pass through?