The information concerning France is NOT accurate. Non TGVs typically have hooks to hang a bike wheel, with a grooved area along the wall for the other wheel to be placed into to keep it stationary. You load your bike yourself and it is free. A bicycle insignia on the outside of the car indicates that a rail car has such hooks for bikes. I took these regional trains several times during a trip to France just 2 months ago. Couldn't have been easier.

As for TGVs, on some routes you can bring a bike, and you must pay a 10 euro fee. Most TGV routes don't accept bikes, however. You can check the SNCF website for a particular route. I took a TGV this year on a route (Paris->Avignon) which didn't accept bikes, but I have a folding bike so it wasn't a problem for me.

I would urge you to not follow the Mediterranean from Nice to Genoa. The traffic is awful. The Italian Riviera is totally built-up and very unpleasant for cycling.

The nicer parts of Italy for cycling are, alas, the hillier parts of the country. I didn't find the Po valley particularly interesting compared to the hills and mountains of Italy. The Adriatic coast is better than the French border-Genoa coastline, but not fabulous for cycling, either.

Similarly, I'm not that keen on the Rhone valley of France scenery-wise, though there at least, you can find quiet roads if you look carefully on maps.

If you don't want to cross an Alpine pass, you can put your bike on a shuttle train under the Simplon Pass connecting the upper Rhone valley in Switzerland with Italy. It takes about 20 minutes for the ride from Brig (CH) to Iselle (I).

There is a large network of bike routes, paths, and canal paths in Burgundy which might appeal to you since they offer flat cycling with pretty scenery in a non-flat landscape.