Best place to rent a car for a driving trip to Giverny, Chartres, Amboise, etc.

If you have 3 days you can remove "etc." from the title. You can do Paris - Giverny - Chartres in a day with an overnight stop in Chartres, then 2 nights in Amboise, but that's about it and it is already a stretch. Maybe consider Fontainebleau instead of Amboise? It is nearer Paris, and it will let you visit Fontainebleau & Vaux le Vicomte (opening times permitting), which are definitely on par with Loire valley castles from a sightseeing poin of view.

I would try to rent a car in the Porte Maillot area (where Avis and Hertz are present) or Porte de Saint Cloud or La Defense. That way you are close to the start of the highways to Giverny.
Then, if you stick to Amboise, I would then return the car at St Pierre des Corps, which is near Tours, and take the TGV train back to Paris: it takes less than 1 hour vs. 3 hours by road.
If you go for Fontainebleau, return the car at Gare de Lyon.

I can't say "best" with any authority, but I have always preferred to take a train out of Paris, and then rent a car.

For our trip this summer, we are going to take a train out to Mantes-La-Jolie, walk across the street to the Europcar office, and rent from there to go to Giverny and Normandy.

How many nights are you staying in France? You would need at least 7 to 8 nights to complete such an itinerary. The basics of Paris require 3 days/4 nights. I like driving but Giverny is easily reached by train from Paris. It makes sense to rent a car to visit but not if your next destination is Amboise and you have very limited time. Include Giverny as part of your Paris stay.

You can also visit the Loire Valley by taking the TGV to Tours if time is limited. It is possible to take an early train and be at the Tours Office de Tourisme (which is just across the street from the Tours Centre train station) in time for the 09h00 minibus tours to vineyards and/or châteaux.

Available time is the critical factor in determining where you can go and how to arrange it.

I suggest that you look at your three destinations in Google Maps (you can even make a route of continuous driving, with effort). The point is to understand that your three-day proposal is, simply, impossible. Giverny is often done by train from a bed in Paris, and people do Chartres as a day out from Paris. I understand that you imagine you are cutting out the "overhead" of those two days out from Paris. But driving around and changing hotels and finding parking at the attractions is not as quick and easy as you imagine.

Time of year makes a difference, since all these places are jammed from May to September. Plus, there are special events (like chateaux candle-light and son-et-lumiere lightings) that might affect your exact timing.

To answer your question more directly (have you ever been to Europe or France before, please?), many people take the TGV to St. Pierre des Corps and rent a car there. They then see the East or West (from there) chateaux, or if they have several days, chateaux on both sides of Tours.

As a guide to your overall plan, we slept three nights in Chinon (TGV from the CDG airport station, rental in St. P des C), three nights in Dinard, 3 nights in Bayeux, and 5 nights in Paris (including 14 Juillet.) That's a much more ambitious itinerary than yours, but maybe you didn't even think that you were missing the D-Day beaches, the only thing people are going to ask you about at cocktail parties! France is a big place.

Thanks for the helpful info. To answer Tim's question:
We are going in early May. Plan was to do 3 nights in Paris, then rent a car and visit Giverny, spend the night in Chartres, then drive to Amboise, see a castle or two and spend the night, drive to Mont-St. Michel and spend the night, and then drive back to Paris for 3 more nights there. On Rick's maps it looked like it would be 2-3 hour drive between each of these towns. But maybe not, it seems like you are saying.

Yes, I travel in Europe every couple years, but mostly in Scandinavia, (relatives) and Germany. We did spend two nights in Paris once, and I was frustrated that I didn't get to see enough.

So this is too much driving? We have nine nights in France. I was hoping to get out in the country a bit.