I am planning a trip to Italy where I will be spending 10 days in the country. I am flying into Venice and would like to also spend time in Florence, Rome and Capri. How many days should i spend in each place, what is the best way to travel from one place to another and if I end in Capri where should I fly out of to come back to the US?

With only 10 days you will be very pushed to just see the big 3 art cities---do not try 4 destinations in that time.
Fly into VCE and home from FCO and take the rails between the 3 cities. Have fun !

Fly out of Rome, skip Capri.

Venice Florence Rome save Capri for another trip. If you take the train you will see a lot of the countryside as you go. I would do 4 days Venice leave in late afternoon to Florence evening in Florence, 2 full days and leave on the 6am train for Rome you'll get there about noon if you take the cheap local train or 2+ hours on the EStar at twice the cost. You'll get to Rome around noon and have several days there. That way you will get a good flavor of each and when you return to do Capri you will know where you want to spend more time. JMHO

In 10 days even 3 destinations is pushing it - I would limit it to 2 - but if you rush and skim you can see a little of 3 places. These 4 just arent logistically possible - you would be spending half your time checking in and out of hotel and riding around on trains.

And is your 10 days 10 full days in the round - or are you counting the days you arrive and leave - in which case you really only have 8 full days on the ground. (We always organize the vacation by where we are spending the night - not counting multiple cities for one day.)