I started family sharing and shared apple music but my kids are still having trouble getting full use of streaming it. Any suggestions?

Then it sounds like they are in the wrong place.

If its suggesting the purchase they are not in the Music App. More likely they are in the iTunes Store App.

To access the Apple Music Catalog, it must be done through the "Browse" and "For You" sections of the Music App on iOS devices and macOS Catalina, or the iTunes application in Macs running earlier versions of macOS.

iTunes for Mac (Mojave or earlier):

macOS Catalina (sidebar)

iOS devices:

What trouble are they having exactly?

Once you invite them, and they accept, they should be able to simple access the For You and Browse sections of the Music app on their devices to stream and download Apple Music catalog tracks and albums.

What happens when they try to do so?