Florence/Chiavari itinerary questions

I am not familiar with Chiavari, but I do know a bit about Liguria & Tuscany, and I can help with your other questions.

By all means proceed by train to Firenze upon arrival in Pisa, unless your flight arrives terribly late. It's a short trip (about an hour) and worth it to settle in your first night. There are about 3 trains per hour, so good capacity. Do yourself a favor in Firenze though and take a taxi to your apartment so you don't have to struggle to find it.

We are very much onboard with car-free travel unless a car is completely necessary. In rural Tuscany, unless you take a guided tour or use an intercity bus, it's pretty necessary to have a car. Depending on your plans while in Chiavari, you certainly could take a couple of nights in a nice town like Volterra, Arezzo, Greve-in-Chianti, or (my favorite) Montalcino. It is an easy-enough car trip. Firenze to, say, Montalcino is only a couple of hours, then Montalcino to Chiavari another 3.5, right up the coast. Returning to Pisa to fly home is less than 1.5 hours, so unless your flight is ungodly early, you can probably drive in from Chiavari the last morning.

Laurel,
Thank you for your input. I will look into Montalcino!

I'm glad to know the train is a good idea. I will utilize a taxi upon arrival to Florence.

Do you have relatives in Chiavari? Together with Lavagna, next to it, it's a somewhat larger than average beach resort town with not much charm. Of course that is my opinion. You don't specify the time you are going there, but in summer (especially July/August) it's busy on the riviera, and Lavagna/Chiavari have quite a bit of traffic. No matter which town you choose on the Ligurian riviera, in summer you are better off with the train.
If I had to choose a charmer town in the area, I would probably choose one of the Cinque Terre villages (which are gorgeous, however get busy in summer), or Levanto, or Bonassola, or Moneglia. I used to spend some time in summer in Moneglia. It's very pretty, about half way between the Cinque Terre and Portofino, and if you absolutely need to see Chiavari, you are only minutes away by train.
The only place where a car would be advisable is the small villages of Tuscany. If you decide to rent a car for a few days for that purpose, it's a good idea to transfer to a hotel outside of Florence for those days in which you have a car, or at least stay at a hotel outside the ancient city walls (traffic is restricted to residents only inside the historical center).

Transferring from PSA airport to Florence (80 km=50 miles) doesn't have to be done by train. I find the luxury shuttle buses more comfortable and once you account for the time on the shuttle from the airport to the Pisa station, the total duration of the trip is similar (about 70 min). Also with the buses, you don't need to worry about carrying the suitcases aboard the train and off the train. You give them to the driver, he gives you a claim ticket, like on the plane, and the suitcases go in the belly of the bus. If you are interested on using the buses from the airport, there are two companies operating them (they depart from the same place at the airport at slightly different schedules. One is called Terravision, the other is called Autostradale. Just google them. They will drop you off in front of the train station in Florence. From there take a taxi to your apartment. If you tell me the address I'll tell you how far it is from the station.

Roberto,
Our trip is planned for late April, early May. The reason for staying in Chiavari is to stay in a villa that neighbors have stayed in previously. It looks lovely and quaint and a nice owner. No particular attachment other than that. This would almost certainly require a car as it is a mile or so up the hill from town.

I will check into the shuttles you recommended. I do not know the specific address yet in Florence. Just the proximity. Supposedly an 8-10 minute walk from the Duomo, but I would utilize a taxi upon arrival to Florence.