A "cheap road bike" or even an expensive one isn't suitable.

Something like a trekking, hybrid bike will do the job. If you live in Europe check out Decathlon. They're a french company but have stores in Spain and the UK as well. Something like the Riverside 5 trekking bike for €400 comes with a rear rack so all you'll need is pannier bags, a handlebar bag, and a couple of straps to tie on your camping gear.

Three weeks should be enough time to do this trip. In France keep to the secondary 'D' roads.

I've never used trains, always planes, so can't really advise.

Check out Crazyguy and Lonelyplanet do a 'cycling france' guide.

I've been to Venice (not on a bike)but at Easter. Believe it's very hot and crowded in summer. Definitely worth a visit but not sure it's really a summer bike trip destination.