13 days in Europe >Italy>Salzburg>Prague

You have too many locations with very long travel times between Venice/Salzburg/Prague.

Your trip would flow better this way.

Rome, 3/Prague, 3/ Venice, 2/Florence, 3/Rome, 1.

You could take a day trip to CT from Florence but would make for a very long day.

I always try to hit my most distant city first.

If you have not purchased air, see if a multi city ticket.. Into Prague out of Rome would be more cost effective. Itinerary would start in Prague with all Rome days at the end.

Edit allowing a nite in the CT.

claire it will take up 1/2 to one whole day every time you change locations. It takes time to get from hotel to rail station/airport, waiting for train/plane, and then getting to next hotel. Add restroom breaks, food, buying postcards, getting directions, cabs etc., and your time gets eaten up. So every place you have "2 days" on your itinerary, consider it to be only one or 1 1/2 actual day of being there. That makes each of those too short to see much. The more cities you try to see, the less you will actually see of them.

As Gerri said, definitely book multi-city air if you haven't already. No point in wasting time backtracking.

To avoid loosing time (and money) by zigzagging and backtracking fly so called open jaw (multi-city). If you go in spring start with Rome (as it gets hotter you are moving north). If you go in fall start in Prague (as it gets colder you are moving south. If you start in Rome to minimize zigzagging go Rome - Cinque Terre - Florence - Venice and vice-versa going opposite direction.

Yes, fly multi-city instead of round trip. That gives you one more night to put somewhere.

Unfortunately I have to fly round trip. I found a round trip ticket for $440 from Minneapolis, MN and multi-city flights couldn't even come close in price.

At this point, I am thinking I will cut Salzburg.

If you decide to spend a night in the Cinque Terre (wouldn't be my choice), go there from Rome, then Florence, then Venice to save a little backtracking.

Here's what I think is not just doable, but will allow you to enjoy a little time in each city.
Day 1. Land in Rome, train to Florence (3N, 2 days)
Day 4. Train to Venice (3N, 2 full days)
Day 7. Fly to Prague (3 N, 2 full days)
Day 10. Fly to Rome (4N, 3 full days)
Day 14. Fly home

Honestly, I would drop Prague and spend all my time in Italy. There are plenty of places to see and you'd use less time travelling and have more time to enjoy. Less is more. On your next trip, see Prague along with Vienna, Salzburg and Budapest.

I think with your time-frame and that you arrive and leave through Rome, that Prague and Salzburg are too much. You will be better served to spend your time in Italy.