Ciao Gordon,
Sounds like a good itinerary!
From the Eliseo hotel website, this is the info they give on how to get to them from the airport (Da Vinci is also known as Fiumicino).
from Fiumicino airport, located 35 Km far from Rome west side, you can get to the hotel:
· by Special Train Leonardo Express
The "Leonardo Express" train leaves every day from Fiumicino Airport station every 30 minutes and it stops at the Termini Railway Station without no mid-journey stops.
· by Taxi
Taxis are available in the arrivals areas of the terminals. The fare to the centre of Rome is about 45 Euro. It is highly recommended to take only the white licensed taxis.
The Leonardo Express has cost of 14 euro per person. Check schedules on the
Trenitalia website here.
You definitely can do a day trip from Rome up to Pisa, spending half a day there and then proceeding to Bagno a Ripoli, which is right outside of Florence.
I don't know Lazio very well so all I can recommend is for places to stop at on your drive up to Pisa once you're in Tuscany: I'd recommend a quick stop in Orbetello to get a view of the lagoon and then another in Castiglione della Pescaia. That way you get to see the Tuscan coast in what is known as the Maremma area.
You never say whether you've already been to Florence. If you haven't then of course you need to spend at least a day here. If you've already been and want to skip entirely, I'd recommend heading into Chianti one day, the second heading toward San Gimignano and Volterra. In Chianti, take the Chiantigiana through Greve, Panzano to Castellina, then head back toward Panzano and take the road to Radda. All this are has beautiful views of vineyards, farmhouses, cypresses. You can also pull out of the main roads and take smaller unpaved ones to get better views as well. Around Radda there are beautiful wine estates, if you head to them you'll find many opportunities to stop for great landscapes. Take the road to Badia di Passignano near Greve or to Volpaia near Radda or Badia a Coltibuono near Gaiole. All great roads with good views, most of them with very little traffic. Even the main roads, being two lane roads, don't have that much traffic - unless you find a really slow car with a line of cars after them. Just stop, take some pics and let them get further ahead
In this period, most small towns have food festivals called sagre but these are often only on the weekends. On your first stop in Chianti, likely in Greve, stop and ask at the local tourism office as they are the ones with the most information on local events taking place - many of these local sagre are not publicized online, only with posters having around town!
Weather this weekend will get a bit cooler here as well but next week's still seems to be pretty good. We've had a really hot September and warm start to October this year. I love it!
Have a great trip and loads of fun!