I spent two nights in Perigueux on a car-less trip through the Dordogne area in 2019. I was hopping from town to town so I could make day trips to some smaller places accessible only by bus. I liked Perigueux.
Most people here rent a car when visiting the Dordogne and focus on the smaller places, but you have to accept that they can be very difficult to reach without a car. I'd recommend checking the SNCF train schedule to see what destinations are practical from Perigueux. Then you'll need to dig into the bus schedules. Those buses do not necessarily run frequently at all.
These are places I went and liked, but I stayed in a lot of different places to expand my reach and don't know what will work from Periguex:
Bergerac
Brive-la-Gaillarde
Sarlat-la-Caneda (big market Wed, huge market Sat)
Cahors
Figeac
Brantome is a small place reachable by bus from Perigueux. If you want to see the market, go early. There was really nothing there when I arrived just after noon.
Martel is accessible from Brive but not necessarily a viable day trip from Perigueux.
St-Cirq-Lapopie is accessible from Cahors with a significant walk from the bus stop, but I don't know that it's practical as a day trip from Perigueux.
Do not attempt Rocamadour under any circumstances; it is a very, very, very long walk from the nearest bus stop along a highway with no shoulder.
I think you would probably do better to split your week across two bases. So many places can be reached only by bus, and to catch those buses you need to start in the right large town/city.
In addition to the Dordogne towns, I'm a big fan of Limoges. That city has a very pretty medieval center and several excellent decorative-art museums. It is also a good base for a bus trip to Oradour-sur-Glane, site of a Nazi atrocity that has a very good memorial/museum. I mention this because Limoges can be reached from Perigueux in less than 90 minutes by train. Personally, I wouldn't want to be limited by day-tripping to Limoges, but that would be long enough to see the historic center and one of the museums.
