Cycling in Europe is a lot of fun. There is lots to see. And usually there are well laid out cycling routes no matter which country you go to. Each country also has a so-called Cycling Association which is similar to the Automobile Association of America which can give you a lot of tips for starters. This is particularly so in the top cycling nations ... Germany, Benelux and Denmark.
The countries you have chosen (France, Italy, Spain) are fantastic places to visit, however may be better suited for more experienced cyclists.
1) vast distances involved with nothing to see ... unless you plan to city hop using the train. Most trains permit bicycles.
2) outside urban areas, don't expect to find English speakers
3) no countrywide cycling infrastructure, but ample quiet country side roads well suited for cycling
4) incredibly hot time of the year and don't expect A/C's anywhere.
5) if you only plan for regional cycling rather than cross-country, e.g. only Paris and the Loire valley, then you are likely to encounter a much better cycling infrastructure
Countries like Germany, Benelux and Denmark are perfectly suited for less experienced cyclists:
a) well established cycling paths, well signed out, throughout the countries

excellent cycling infrastructure, including cycling motels and repair shops all over the place
c) the urban density is higher than in the southern countries, which means more to see every few miles and more English spoken
d) not so incredibly hot in the summers
Eric