Between Chinon and Amboise, I'd choose Chinon! We didn't get to explore the town of Chinon -- just spent our time at the fortress, which is beautifully restored and full of informative signs and audiovisual effects -- but the RS guide to France describes a self-guided walking tour of the old town that I would have loved to do. The town is in a picturesque location on the river Vienne, a Loire tributary, and it's surrounded by vineyards. We bought a bottle of rose wine which we greatly enjoyed later in our trip.
About 45 min leisurely drive from Chinon along a straight road is Azay-le-Rideau, which has its own chateau and several high-end restaurants as well as more casual places to eat. We stayed there in the Hotel Biencourt and it was wonderful. The innkeepers are the nicest people you'd ever want to meet. However, if you choose to stay in Chinon I'm sure there are nice people there too!
We spent a short time in Amboise on our way to the Chateau of Chenonceaux and found it be to comparatively crowded with tourists on foot, on bicycles, and in cars -- all acting kind of aggressive about wanting the right of way in the narrow village streets. My impression is that once you get to the western/southern side of Tours, the places that are out of range for an easy day trip from Paris are more relaxed and more characteristically French.
We would have loved to venture further west to Villandry, Fontevraud Abbey and maybe even Saumur, but lacked the time. Next trip!