Well..... April this year has Easter (April 5th) and while you can just wing it for some parts, I would get those first nights booked. Not knowing exactly when you arrive, if it is around Easter, you should definitely book as soon as you can. April 25 is a national holiday so I'd also book ahead around those dates.
I would suggest Florence first since you arrive here, then you can most definitely train to Lucca for your week there. If you have the time in the itinerary before Lucca, definitely a day in Cinque Terre would be perfect as it is to the north of Lucca and then you can concentrate on heading south. You can do Pisa while in Lucca as a day trip.
Once you leave Lucca, I'd definitely suggest getting the car and driving toward Chianti and Siena. You can also decide to find another base for a week and use that to roam around, if you're central in Chianti anywhere between San Gimignano and the towns in Chianti (Radda or Panzano or Castellina, for example), it will be a good base from which to set out each day and explore. You can also Then spend a few more days at another base south of Siena - many hotels and B&Bs do have last minute availability and many are on the major booking platforms that make them easy to find while you're here, but if you know you want to stay somewhere for a few days, booking ahead is generally better so that you don't have to move out every morning.
In the countryside, it is likely having the car won't be a problem at all, most places are not in the centers of towns but outside. So you should have parking included and will make it easy for you to move around and explore. Most small towns have parking areas outside of the historical centers, so look for the "P" signs as you arrive in towns and head there. Most of the centers are closed to traffic and are pedestrian only areas generally, which makes them more enjoyable while visiting on foot.