You can do it, although most of the relaxing your friend is looking for will be happening on trains and in train stations. Travel times are somewhat deceptive; the schedule says 2 hours by fast train to go from Florence to Venice, but its not the case that you'll be touring Florence until mid-morning and be touring Venice by lunchtime. You need to pack, check out of your hotel, get to the train station in time to get on the train, and then on arrival find a new hotel. Even with reservations you still need to find the new hotel and check in, drop off your bags, etc. which itself is easily an hour.
My personal experience is that even a 2-hour train trip really takes all your time from waking in the morning until around noon or 1pm. So your trip may consist of arrival in a city in the afternoon, the rest of that day and the next full day in the city, then repeat. If you're ok with that then the trip is doable. If it was me, I'd drop one of the cities.
June is a nice month to travel in Italy, but it means lots of other tourists will be there too. I've traveled without reservations and with them; the advantage of going without is that if you want an extra day somewhere you just do it, if you decide to move on then you can go. The price is twofold; the cheaper accomodations may be taken, and you spend your valuable time looking for a place to stay. Given the short time you have, I'd suggest reservations.
Lastly, as RS says cars are an expensive headache in major cities. Each of the places you mention is well served by rail, and a car won't work that well in Venice unless it floats :-) You don't want to be driving a car around Rome, and in Florence you may end up getting a few of the expensive traffic tickets for driving in resident-only zones (see http://www.bella-toscana.com/traffic_violations_italy.htm)