wbfey1 said what I wanted to say but I'm afraid that I'm often seen as negative and didn't want to share the negatives of the drive contemplated.
That said, now that the pussycat is out of the cloth receptacle, the description I use for the road between Genova (Genoa) and the French border is a Venetian Blind. It really truly is tunnel after tunnel after tunnel after tunnel, and neither the inside the tunnel road nor the bits between the tunnels can be described as anything like straight. Narrow too. The highway that just fell down was in Genova.
Tolls in Italy are high, there are speed cameras, fuel is the most expensive in Europe, they have TUTOR, there are ZTLs with delayed high fines, and IDPs are required. If any of those abbreviations are unknown it is worth looking them up or asking. Parking is very expensive or non-existent.
Tolls in France are extremely high (in the range of €90 to €100 each way from north to south (I saw each way because of the cost of repatriating the car will include it for the return journey), speed cameras are everywhere, fuel is very expensive (that's one of the reasons for last week's riots), IDPs are required (or at least are the cheapest and easiest way of meeting French law), and parking in Nice is extremely difficult. When I visit there I often marvel at how many cars have dents from parking.
I understand that appeal of a road trip - I've done my share, and I drive to Europe several times a year. The trip you are contemplating has a few hazards however, and the route you will have to take won't be scenic.
Figuring out the trains can, with a little information on how to do it, can be very straightforward and easy.