My daughter and her boyfriend are doing a similar northern Italy combo. flying into Venice for a few days, then spending a few days in Parma to do day trips (bologna) and a food tour. Then going to Cinque Terre and leaving from Milan. They actually picked this because it had the best airfares as compared to just about anywhere in Europe at the time they booked. They are going in early September.

OP is thinking of May or early July - May will be a LOT nicer, cooler weather and a lot less crowds at tourist sights.

July only if can't go other times IMO.

It all ends up coming down to personal preferences.
After my first trip to Italy, I decided I would (ideally) love to retire there. A friend of mine went to Italy and said she's no desire to stay for any length of time and claimed she had the worst food. I, on the other hand, believe I had the best food, and met some of the loveliest people. They went to tourist restaurants, I went to where the locals hung out. It's all what you prefer. If you want to really get into a place and the feel for it, go off the beaten path when you can and find the locals. If you want to cram as much to do in a short amount of time, then follow the main stream tourists. I tend to do a little of both depending on my mood and location.
I've been to Paris twice, and wouldn't care if I never went back. The countryside in France and the people outside Paris were lovely and I would return to see more of that side of France though there are other places that are above it on my list.
I'm a history junky and love to find at least one bizarre bit on every trip that most people have forgotten about. I'm sure you see what I'm getting at.
Ask yourself a few questions. What do you love to do (hobbies, food, museums, shopping, art, music, photography, etc?) and make a list. Then do some research into the areas you're considering and match them up. These lists will change from one year to the next as change is inevitable. And always travel with the thought that you will be back. It takes the pressure off of having to see so much. I am currently trying to sort out my trip to Puglia and there is SO MUCH to see there, I had no idea. I thought I'd be bored with trying to come up with some things. Not so much. Now I've decided I will have to return as there is no way I will get it all done and enjoy it at the same time. So I'm whittling away my list so that I can see some fab things but take the time to enjoy them as well, and leave space for random events/places as well. In the end fate will have the last laugh and I/you'll end up having experiences you didn't think, or want, but will just roll with it and embrace it all in the moment.
Happy travels! Enjoy and just be present in the moment and it will all come together and you'll have a trip that you'll always remember.