I've spent a lot of time in Italy and have never been the victim of petty crime. As long as you behave with the kind of standard caution you would also use in New York City, your chances of falling victim are about the same as they would be there.
On running in Florence, it depends a bit on just how far you wish to go. There are some truly wonderful trails north of Florence so if you want a run of more than 10M or so, I would strongly recommend finding how to get to those. There is a map of the trails available in most bookstores in Florence. I can look up the precise title of the map at home if you wish (just drop me an email.
If you want to run shorter, running along the Arno, heading East, you can go for a couple of miles alternating grass, gravel, and old flagstones. Very pleasant.
If you want good views, take the bus 7 up the hill to Fiesole and run around there. By the way, bus 7 is allegedly notorious for its pickpockets, so do be extra vigilant.
In Rome, if nobody has suggested it yet, I strongly recommend Villa Pamphili -- the best place to run in all of Rome.