Yes, you can do it! With limited time AND budget, I decided I was going to make Italy a reality for me. So I literally googled “Italy in a week”, and lo and behold, an article came up. The guidance, hit three of the most frequented cities- Venice, Florence, and Rome. If you start in Venice, then travel via train to Florence, then another train to Rome. You can also do this the other way around- fly to Rome then take a train to Florence then a train to Venice.
Venice- Surrounded by water on all sides, you have to travel here by boat after you fly or train it here. I suggest booking a Vaporetto (water taxi) beforehand to come meet you at the airport. Here is a Vaporetto. The ride via Vaporetto is gorgeous. You will pass many tiny islands that are out of a movie before you dock in Venice. This is the type of thing you will see.
Yes, Venice is gorgeous. There is history and mystery here, and if you find a small hotel where you will get a local feeling, be prepared to walk through what seems like a maze of shops and buildings to find it. This is an island of mostly tourists, very few Venetians actually live here. You can get a local experience by stopping at one of the small gelato stores and getting some seriously amazing gelato.
There is must see island off Venice- the colorful island of Burano. The houses there are all painted bright colors and are super adorable and quaint. My suggestion is to take a Vaporetto from Venice which should take about 10 minutes and have lunch at an amazing local restaurant called Al Gatto Nero Da Ruggero. Order the Risotto and enjoy an amazing local and delicious experience.