You probably already know this, but for the flight make sure you get up every couple hours and walk around, in your seat do leg/ankle exercises.

The itinerary you propose is doable in 16 days but that would be a minimum and you would need to be selective about things. Since the single biggest cost is the airfare, once you are over there extending by a few (or several days) is a good idea. I think for a first trip 5 days Amalfi Coast, 5 days Rome, 3 or 4 Venice. Florence itself 'deserves' at least 2 or 3 but there are numerous day trips you can take from there - Pisa/Lucca by train one day, Siena by bus another day, a day tour to wine villages, etc. If you want to base in a smaller town that's possible, but usually best done if you have a car and except for those few days in Tuscany a car is not needed (and can be a pain). I would base in either Florence (or Florence and Siena) and see a few smaller towns by taking a one day bus tour. Three weeks would be a nice trip. I regularly take 5 week trips now and love it, but for first timers I think three is better as you don't know yet how you'll feel.

Definitely only take one carry on (21 inch or less) bag and one personal item - tote bag or back back. They don't usually have strict sizes for the personal item as long as it can fit under the seat in front of you, so I pack quite a lot of stuff into that. I sometimes don't even do 'actual' laundry. I do underwear and even tee shirts in the sink of the hotel. You might also say I lower my standards of how 'clean' my clothes need to be when traveling. Dark colors at least for the pants/skirts.

If taking trains or buses - number one rule - PACK LIGHT - lots of stairs and narrow passageways to navigate once on the train. And have wheels - sometimes long walks in stations.

Train and plane travel is much safer than bus for DVT risk. You can walk in the aisle or just stand up without too much difficulty compared to a bus.

Look into "packages" of flight plus a few days hotel at beginning or end or both . Could save money without limiting independence.

How we limit packed "stuff" -- Daily shirt lasts 2 days instead of one. Travel days use a shirt for third day. No one cares what you look like. Clothes/underwear that have one more useful wearing is packed and discarded after last use leaving room for purchases. If you are tempted to pack something "just in case," don't. If it doesn't happen you are home free, but if you do need it, you are in civilization and can buy anything you discover you need.

Here are some things to consider, that have worked for my parents (older than you) when they traveled to Europe on trips I helped them plan, and one that I had the special delight of taking with them. Since you're saving a bit of money on not taking a tour, consider taking some small-group tours in each place, including some food-related tours. We actually took a couple of private tours, because we could ask the guide a lot of questions about their lives. Arrange for "transfers" to and from lodging and airports and/or train stations. Especially that jet-lagged arrival day; it would be so nice to have a person with a sign waiting for you in the arrivals area, who will take at least one bag to their vehicle.

However, If you fly into Venice, consider splurging (if it fits your budget) on a water taxi from the airport to your hotel. I think it's one of life's greatest pleasures (and I'm someone who's pretty economical with my money, most of the time). I even enjoyed it more than a gondola ride.

For any museums that you KNOW that you want to visit, purchase tickets online, in advance. Many of them do not require a certain timed arrival, though some do, but it makes it so much nicer to be able to skip the line. You can even do this with the Vatican Museums.

"Schedule" a lot of people-watching-at-a-cafe time. Wine, coffee, fizzy water, whatever.