Rent a car or train in the outskirt of florence?

Do you mean Villa Olmi in Bagno a Ripoli? If so, then renting a car for both trips might be worthwhile. The buses from Bagno a Ripoli go to San Marco rather than the train station, so you'll have to take an additional bus to get there (many options). Having a car will allow you to return directly to you lodgings after visiting Pisa and Lucca and not spend time renting one on Monday. However, you'll need to check which rental car locations are open on Sunday as at least some will be closed. In Pisa there's a large parking lot near the Field of Miracles that the tour buses also use. Parking outside the walls in Lucca is best. Be careful about sampling wine and driving on Monday, as legal limits in Italy are low.

A car would not be necessary for Pisa and Lucca; nor SG, which is reached from Florence by a train or bus and a bus. Note that SG gets hordes of tour bus visitors.

Villa Olmni is a farmhouse 8.5 kms from the SM Novella station in Florence, to which they have a free shuttle; missing that on the way back you would take a taxi. Train time to Pisa is one hour; Pisa to Lucca 30 minutes. Combining Lucca IMO is a mistake, because it means adding 2 hours on trains and going to and from stations. Lucca better a separate trip and there are others. Pisa there is a lot to see:

https://wikitravel.org/en/Pisa

And here are seven self-guided walks:

https://www.gpsmycity.com/gps-tour-guides/pisa-4960.html

Siena is also a very worthwhile day trip from Florence; the bus which goes to the center of town.

To rent a car [on any day] you would need to take a taxi from the SM Novella station to Via Palagio degli Spini, the complex out near the airport. Because of the time involved before driving anywhere and back, a one-day rental is not efficient.

Also remember if you drive to wineries and imbibe, drink driving is has much stricter limits than North America (your profile doesn't say where you are from).

There were similar answers to your previous post here in March - for reference it was https://community.ricksteves.com/travel-forum/italy/do-i-rent-a-car-to-do-day-tours-from-florence

Good point. The limit is .05.

Villa Olmi is near the neighborhood where I grew up. My high school (Gobetti high school) is across the street.

It’s an ancient noblemen’s villa which was converted into a hotel only recently (less than 15 years ago). It’s a gorgeous place to stay and also ideally located for day trips to the Chianti and anywhere south of Florence. The Chianti Road starts right there and so does the entrance to the A1 freeway. I’ll be in the neighborhood in July also visiting family and friends.

You need a car to stay there. The closest bus is the no. 23, which is frequent and takes you downtown (including to the Santa Maria Novella train station) with no changes. However the villa has a very long driveway, then you need to cross a busy boulevard (Via Del Pian di Ripoli) and take via degli Olmi to via di Ripoli where the 23 bus stop is. From the hotel lobby to the bus stop is a half mile walk which I wouldn’t want to do in the scorching July heat. To visit Florence itself or to go to the station, either take a taxi or drive your car to nearby Piazza Francia behind the Italiana Hotel, park there and take the 23 bus from there. For Lucca and Pisa you could take the train, but since you have a car, I’d drive. You are next to the freeway entrance anyhow. I can guarantee you that in the time it takes to go from your hotel to the Florence station, you can almost be in Pisa. From there to Pisa it would take me 50 minutes. I could pickup people at the Pisa airport late in the morning and be back home by lunch.
Any other location you mentioned that you want to visit is best done by car.

While there you have a great restaurant to go to, just 10 minutes walk from your hotel. It’s called La Bettolina, in the Nave a Rovezzano district along the river. It’s owned by an Albanian family (it’s Italian cuisine though). They specialize in seafood. The best in the world. If you go it’s faster to walk. It’s through some amcient narrow country roads surrounded by villas and farmhouses. They are one way streets so if you drove you have to go around, it’s easier to walk. However take flashlights with you at night because they are narrow with no sidewalk so you want to signal your presence to the rare cars which might drive by. The area is very safe. We used to steal cherries from the farms there when I was a kid (we were the only criminals, a baby gang of cherry stealers). Now those farms are all converted into fancy residential villas.

Thank you for the replies. I hired a driver to pick us up at train to go to hotel . I will probably rent a car for the 2 days. The hotel has the car brought to us.

I've driven into Lucca a couple of times with no trouble and parked outside the walls. For Pisa, I made the mistake of not mapping my approach in advance - don't make the same mistake. Just trying to follow signs to "centro" and "stazione" led to a very circuitous route.