Hands down Xavier's apartment is probably one of the best you could hope to find in Lyon. It's located in a safe, high-end neighborhood, surrounded by high-end boutique shops. The area is highly walkable and we walked to everything or just grabbed a couple of Vélo'v bikes (the bike share program with stands all over Lyon). The metro is available as well with a stop in nearby Place Bellecour. You can also use Uber, G7, or I suppose a regular taxi if desired (they will often station themselves in front of nearby hotels).
Grocery shopping is at-hand if you are booking a longer stay. There is a large Monoprix within walking distance but also a mini Monoprix at Place Bellecour and there are Casino shops (mini marts) scattered around all over the place too. Vieux Lyon is literally down the street and across a bridge, maybe a 10-15m walk. Don't miss the Saint-Antoine market along the river practically every day of the week.
If you need to venture further, the main train station (Gare Lyon Part-Dieu) is a bit of a distance and you will need transportation to get there. But you can take a French TGV to Paris or an Italian ‎Frecciarossa to Turin or beyond from this train station -- both are high speed trains that travel in excess of 290km/h. If you are headed to London your closest train may very well be back at the train station next to the airport.
This was a new rental when we stayed and there were a few problems but those should all be resolved for future bookings. Throughout our stay Xavier was always available via chat in the Airbnb app despite the fact that there was also a property management company.