Most things have been said alreday. Just a few more hints and thoughts and one correction. The absolut tourist peak is not June or July, its August. Especially the week with 15 August, which is for the Italian something like the most important day of the year. Everything with legs and wheels is out and running either to the mountains or to the beaches. Most tourists in Italy at this time are Italians. Additional the 15 August is a national holiday for some of the sorrounding countries like Switzerland, Germany and also we Austrians are heading south that day. Many Swiss, Germans and Austrians get to Italy for a long weekend in this specific week. So try to avoid it or be prepared to see crowded mountain roads jammed with trailers, caravans, motorhomes and probably 5 million bicyclists riding side by side making the jams even worse. Only in China I've seen more bycicles....
Best riding area are the Dolomites and northern Italy in general. From there you can go to Austria (very, very nice roads), Slovenia, Croatia which all offer noumerous mountain roads and many places worth to look at closer.
There is much more Police in Austria than in Italy but the fines and fees are far less. Our cops are not as cool as the Italians who just have a heart for fast and loud vehicles of all kind. Always depends if you catch a nice guy or asshole. Very, very good chances to meet nice cops in Italy and Austria, 100% ashole cops in Switzerland and most expensive fines and most severe punishments there. So go slow, these guys have strange laws (a bike is considered as weapon once exceeding certain speeds!!!!). I'd avoid going to Switzerland aslthough they have dreamlike mountains, but are in general not too friendly to foreigners. If you have the chance to visit countries next to Italy I recommend France, Austria and Slovenia and Croatia (although this requires to go through Slovenia).,
Dolomites, for me the best by far but I am an Italy fan since decades so others may see that different. Another good place, one of the best in Europe is Savona. Its further in the west an not too far from French border. If you like gravel roads and endless offroad trails this is the place to go. You can try to ride the Ligurian Mountain Border Road. (check ADVR Forum Riding>Day Trippin> Ligurian Border Ridge Road).
All the best