Going back to your OP, here's a suggestion:

Day 1 travel
Day 2 arrive in Rome and stay until
Day 6 then take train to Spoleto (or similar) and make that your base for Umbria/Ascoli Piceno trips until
Day 11 or 12 take train to Venice and stay until
Day 14 and leave (early) from Venice

You can do a long day trip to Florence from Spoleto (or Rome-there are guided tours). If you have a car, you will save some time but train travel is pretty easy.

Google 'Rome to Florence Tours' and 'Spoleto to Florence Tours'-there are many options.

Have a super time and please report back about how it goes and what you finally decide to do.

I am planning my second trip to Italy and came across this post from early on my planning of the first trip 3 years ago. I thought I would give you a summary of what we decided--the trip report never happened. Arrive Rome, train to Florence. This worked well as we dozed on the train, minimizing jet lag. Florence for 4 nights which was the right amount of time for us. Rented a car at Florence airport--we had international driver's license but missed the part about needing the U.S. License. After some very tense conversations, they allowed us to rent the car without the U.S. Driver's license. Plan was to drive to Perugia, Assisi and Then drive to Ascoli. We spent too much time in Perugia, and drove direct to Ascoli. Had an amazing visit in Ascoli with family and a large family trip back to Assisi. 3 nights at Villa Chicci. Car to Fumicino and then taxi to Rome, 3 days in Rome--not enough but a good start. Thanks to all for the great advice. Upcoming trip--start in Venice, drive to Emilia Romagna,down to San Marino, back to Ascoli and then....