The big question for your predicament is budget. Remember that every time you move, you’re costing yourself money… SO, if you can fork over the cash and you’re willing to pay to see everything, I say go for it! Your first plan with all of the Italian cities PLUS Greek Islands is a lot to handle for 10 days.
The great thing about Italy is that it’s very accessible by train. If you wanted to do all of the cities you could do something like this: 2 days (one over night) in Rome, one day/night in Florence with an early train to Venice for the day and then back to Rome for a night. Then two nights at the Almafi Coast which leaves you 4-5 nights for 2 Greek Islands. That is the super deluxe trip for the super ambitious traveler. It’s doable but you’ll be in trains and on ferries more than you’ll be on a beach or in a vineyard so I’d suggest something more like this…
Based on what you said earlier, it seems like you’re less interested in tourist stuff and more interested in relaxing into a country. You definitely can’t cram everything in, so instead, choose different destinations so you can get a multifaceted trip.
Start in Rome and spend one night here. Check out the Trevi Fountain, Spanish Steps, the Forum and wander through the streets during the day. Take advantage of the Colosseum Pub Crawl and other great night life opportunities at night. The next morning catch a train from Termini Station in central Rome to Santa Maria Novella in Florence.
For wine, great markets/shopping/leather stores, good food and gorgeous views of the Arno River:
Don’t miss Florence since it IS the heart of the Chianti wine region and one of the coolest Italian cities in my opinion. It’s small enough where you won’t feel rushed like you might in Rome, prices are slightly cheaper than Rome, and wine tasting day trips out to Chianti vineyards are plentiful. If you do a wine tasting tour, spend two nights here. Rome is great, but almost everyone I meet who travels in Italy tells me they wished they spent less time in Rome and more in Florence. Check out PLUS hostel, it’s cheap and it has a great terrace bar—you’ll be sure to meet other travelers and it’s centrally located.
Then take a train south and spend two nights on the Almafi. There are a lot of companies that do a water sports package if you’re interested in checking out the sea caves, sea kayaking, surfing, cliff jumping… the Almafi Coast is going to be very similar to the Greek Islands so shove on to Athens via ferry from Bari or flight.
Skip Athens entirely (come back to see the amazing Greek architecture on another trip though!) and head by boat to any island of your choice. To save time and money, think about booking overnight ferries. They really aren’t that bad. I just went back to Greece a few weeks ago and paid 50 euros for a roundtrip overnight ferry to Santorini. With one night on the island and two nights on the ferry (going there and back) it was perfect. We didn’t waste time being on a boat during the day, but if you want to catch cool pictures of the sunrise you’ll still be on board by 8am so don’t worry! You de-board and you have the whole day ahead of you. I thought going back to Santorini twice in eight months was excessive but I had a totally great second experience that was very different.
My other favorite island was Ios–especially if you’re down to party, lay on the beach, and meet tons of great Australians. If you want more advice on just how to do the Greek Islands check out this post. Don’t forget to rent an ATV!
So, my best advice to get the best of Italy and the Greek Islands for 10 days is:
1 night- Rome
2 nights- Florence
2 nights- Almafi
5 nights- split up between 2 Islands.