I am trying to find a cheap and quick way of getting from either Florence or Venice to Athens in mid July. Can anyone recommend the best way to do this?
Cheers

A ferry Venice to Athens will be slow and expensive.
Visit travel agencies and try to find the cheapest ticket to Athens then compare that with the cost of train-ferry-bus Venice-Brindisi-Corfu-Athens and don't forget to consider the time spent travelling.
A few years ago I was planning to take a ferry Italy-Greece but I got a ticket Rome-Athens for less than what the train-ferry-bus would have cost me. I visited 3 travel agencies and asked about the cheapest ticket Rome-Athens. It turned out a return was less expensive than a one way (logic notwithstanding) and it also turned out a 2 week return was less expensive than a 3 weeks return (SPQR). I actually saved money and time. I simply missed my return flight.
Rule and price structure change all the time so do a bit of leg work.
Also keep in mind a big established airline is not necessarily more expensive than a cheapo airline; while this is generally true, there are exceptions to every rule.

There is a ferry from Ancona to Corfu. Its overnight, if I remember correctly. Once you get to Corfu, you have to take another ferry.
If I were you, I'd fly. Greece is a long, long way from the Europe mainland and flights are probably less expensive

I'd hazard a guess and check skyeurope via Bratislava. This will still cost about 150 € for the two flights if you can get a connection.
Oh curiosuity got me .. eaysjet from a flight from Milan Malpensa for 75€ July 18 .. otherwise you will spend more than than on the train to Brindisi & 60 € for the fery to Patras ..