I think you should look at getting some sort of Railpass, even in Italy it could be cheaper than ordinary tickets. The railways are excellent in Italy, so this should be your way between cities.
I find Venice kind of dull and expensive, I would try to find a cheap ticket to Milan or somewhere nearby and start off from there. Milano is a huge city and not very nice, but it has some interesting sites - the Cathedral, Big-Market-with-ceiling-thing and the Train station.
From Milan you could go almost everywhere! There are some nice cities along the "Riviera de Liguria" with train access. Sestri Levante is my favourite town, but uncertain if there is some sort of Hostel there, it can gete expensive, especially in high season. So, the western coast has the ocean and severeal real italian towns, but fewer big cities (with cheap hostels). The bigger cities in Liguria are Genoa and La Spezia, accommodation may be cheaper there.
If you travel through Italy in August without having some days at the beach you'll porbably swet to death. Rome can be one of Europes hottest cities in my experience.
From the Coast (Liguria), you want to move on to Florence, this is doable with train, but you'll have to stop in Pisa (Maybe not such a bad idea?). I reckon a day in Pisa is more than enough, but i haven't been there since I was 13, so things may have changed.
Florence is a great city! Lots of sites, good shopping and billions of drunk American college students. Plan at least 3-4 days in this HOT city.
Rome is a four(?) hour traintrip from Florence, I know a ok hostel just by the train station, if you are interested.
700£ isn't very much I think, so you should probably shorten your trip by a week or two and not try to see all of it. Fly to Pisa - train to Florence - and train to Rome - Fly home from Rome?