One of your first stops should be Aubusson Visitor Center. Here, you'll be able to find your bearings and pick up some great ideas before you set out exploring.
You'll have sightseeing choices galore when spending time in Aubusson. Aubusson Visitor Center is a good place to start here, but there are loads of other fine places to discover. International City of Aubusson Tapestry, Musee Departemental de la Tapisserie d'Aubusson, and The House of Upholstery are near the tourist information center and and will cap off your day nicely.