Looking for interesting and yet non touristic places in Italy at the beginning of the summer is a hard task. If you are just looking for places less crowded than Florence, the matter simplifies sustantially: there are plenty of nice places which are definitely of touristic relevance but do not get too congested, especially if you avoid the weekend.
In this case I would strongly suggest Lucca (just west of Florence) in Tuscany or Bologna, Ferrara and Parma in Emilia-Romagna, if you are willing to take a deviation towards north. Ferrara especially is a favorite of mine and I remember spending there a pleasant Sunday in June, when I went there to watch a tiebreaker match for Monza soccer team to advance from third to second division. On the shortest path to Cote d`Azur, going through Liguria, you would have to chance to stop by the so called "Cinque terre" (five villages on the sea close to La Spezia: Riomaggiore,Manarola,Vernazza,Corniglia and Monterosso). Admittedly, they are pretty touristic but not as heavily as Florence and not so much out of the weekend, at least before July. The place is stunningly beautiful, and this from a person who strongly preferes hiking on the Alpes to any sort of beach.
If your aversion towards tourists is total, there are also places worth seeing where you won`t find many people: in Liguria, close to Imperia, you have Cervo, a small village on the mountain (a short and nice hike) where you can sip Porto from cafes with view over the coastline. Genoa is also an interesting city (which its characteristic narrow street, became famous in the footage from 2001 G8) and it is not touristic, but it is a big city. Just north of Liguria, there is a nice and not mainstream location in Piemonte, known as "Langhe", which is one of the most prolific areas of production for Italian wine. It is a very quiet countryside with vineyards and stuff. The main town in the area are Millesimo (still in Liguria), Cherasco, Dogliani, La Morra and Bra. If you happen to go there, there are a couple of Medieval castle worth visiting: Castello Barolo and Castello Govone.