If you go by car and the Alpe D'huez is a must, then I'd advise you to take that as last stop in France. It's relatively close to Briançon and from there you can easily pass the border into Italy. If I remember correctly (used briancon to get into Italy 2 years ago) it's a mountain pass and after that you get onto the highway to Turin pretty quickly. Once on the highway you can easily follow it to Milan > Dolomites / or all the way to Venice if you want to go there first.
Regarding to places, I don't really want to know where you want to stay for how long. Loads of nice places in the French Alpes. At Bourg d'Oisans you got the alpe d'Huez, but more interestingly you got the Glandon, Galibier, etc. Also the balcony road Mr.Gibb suggested. I did the Balcony road from Le Freney d'Oissans last year, also great route with spectacular view over the valley, here is that ride (ignore Les Duex Alpes I did before that) https://www.strava.com/activities/625506848
If you like it steep then you can also ride to Vaujany, small climb when your riding from Bourg towards the Glandon, It''s a relatively lesser know climb but it doesn't go under 8/9% on it.
Then there is also Bourg Saint Maurice. Really nice area to ride with the Petit Bernard, Iseran, Cormet Roselend, Col du Madeleine and more close by. Also a place where you can ride into Italy via the Petit Bernard into Aosta. From there it's also easy acces to the highway onto Milan.
There are plenty more nice places in France, I don't know how south you go, else the Mont Ventoux might also be an option. I don't have any experience with the Swiss/French border so can't help you with that, but there are plenty of swiss riders here who probably can.
The dolomites, I'd say stay anyway near Corvara, Alta Badia, etc. There you got acces to a few local loops like the Sella Ronda, you can make a nice loop with Giau, Falzarega, Valparola and many more. The climbs in that region are beautiful.
Personally I'm going back to Canazei in September, not as central position as Corvara, but a bit cheaper and most climbs are still in reach.
If you like it small-town I've also been to Borgo Valsugana, it's in a valley near Trento. I've only done the Passo Manghen there but I believe there are plenty of other nice climbs there.
If you want to go more towards the Italian Alpes you can go to Bormio and ride the Stelvio, Gavia, Mortirollo etc. Also nice region with great rides. 1 of my personal favourites there is Cancano. Not as hard as the others, but I just like riding towards it with the sight of loads of switchbacks.
There are plenty of more places, but hope this helps you a bit!