Italian here. End of July/beginning of August is a TERRIBLE time to be in Rome. It is typically sweltering, many stores are closed for the summer, and it is inundated with tourists. Venice is no fun, either: the canals tend to smell foul when it gets hot.
The trick is to get up early, run around in the morning, go back to your hotel for an extended lunch, and explore some more in the evening into the night. Catch a siesta!
Venice: the most amazing thing about Venice is how ALL the tourists stay on the straight line from sight to sight. Ignore straight lines and you'll be quicker and see a lot more. Also: take the ferry gondola from one side to the other of the Canale Grande. So much cooler (and factors cheaper) than the touristy gondola with the singing gondoliere. They have them at several locations on the Canale, and you see them easily from any bridge. Oh, and Venice is usually deadly boring at night (might be different in summer).
Florence: don't stay in town, explore. Day trips worth taking include Siena-Chianti-Volterra, Assisi-Perugia, Pisa-Portovenere, Bologna-Ravenna. The Tuscan coast is also worth a trip, especially the part South of Livorno. In Florence, the cathedral, the Piazza della Signoria, and the Uffizi are musts.
Positano: Great choice, the whole coast is breathtaking! Make sure you take the ferry to Capri. There, skip the Blue Grotto and take the bus to Anacapri and visit the Villa San Michele.
Rome: say hi to my hometown! If it get too hot, leave town for either the gay beach (41.6572, 12.4142 on Google Maps) or beautiful and cooler Tivoli (see the gardens and Hadrian's Villa - did I mention Hadrian was gay?).
Once you know where you are staying, go to Google Maps and search for caffe, gelateria, ristorante and other ITALIAN words. You will get results in Italian, with the reviews and votes of Italians. Romans know that you don't get the best food and drink inside the walls, where real estate is crazy expensive, but outside.
On that topic, stay away from hotels near the Stazione Termini. Some areas are definitely sketchy. And do not drive a car in Rome without full insurance coverage. Both accident and theft. Trust me.
The mandatory sights in Rome are the Vatican (Saint Peter's probably more than the museum), the Forum and Capitol, the Colosseum and Gay Street (right next to it), Trastevere (especially at night), one of the catacombs (I'd do Priscilla, next to the beautiful park of Villa Ada), E.U.R. in the South (the large scale model of ancient Rome is particularly fascinating). You can't go wrong, though: in any other place, Rome's secondary sights would be a reason by itself to go there.
More in general:
* do not EVER eat at a restaurant that has a bilingual menu or a menu in English (and run as fast as you can if the menu has pictures).
* The coffee bars that are everywhere usually have delicious snacks.
* Try at least once to take the bullet train (Rome-Florence is the easiest connection, but taking the train into Venice is quite amazing). You can buy your tickets in advance on the trenitalia.com site).
* Italians love socializing. If you don't speak any Italian, but learn "mi scusi" (excuse me), "grazie" (thank you), "parla inglese?" (do you speak English?), that's pretty much all you need to get by.
* Make eye contact. It is expected of you. Not making eye contact is considered weird and somewhat suspicious.
* If you can, avoid driving in urban centers, especially South of Florence. The traffic will drive you insane. Also, you will not be used to European freeway speeds.
* If you are on a budget, choose a trattoria over a ristorante. If you like good food, do the same.
* Eat what's in season. Don't ask for artichokes or strawberries in August.
* Ask Google about good pizzerie. Type "pizzeria" to get the Italian reviews.
The advice on drinking bottled water is, with all due respect, horse crap. Rome's tap water, in particular, is better than bottled water. You will find water fountains everywhere in town, and you'll drink the same crisp, refreshing mountain water that the ancient Romans drank.