My female friend and I, both in our sixties, are going for two weeks in mid May. Was considering Lucca as a home base, or should we do a week in Lucca and a week somewhere else? Pisa? Carrara? Or is two weeks in Lucca good, with day trips to other smaller cities. We are flying into Pisa. No desire for big cities. Prefer not to rent a car and drive.
If you prefer to use public transportation then you should stay where the trains stop. Also buses. Consider Pisa, Florence, Certaldo, etc. in addition to Lucca. Check out a Trenitalia map. Most all towns are reachable by train or bus, but it takes longer to get around.
I would break up the two weeks and stay one week in Tuscany, and one week further south in Tuscany, or in Umbria.
Without a car (which is also the way I travel), train and bus links are key--especially when you're contemplating spending a full week in a not-large city/town. We really need to know what places you want to visit.
All the potential bases you mention are in northwestern Tuscany and not centrally located. They have zero or few Freccia (fast) train links to Florence, the city from which rail lines tend to fan out--thus a city where you might well have to change trains to reach destinations in other parts of Tuscany. Pisa, being a quicker trip from Florence, might be more practical than the other two from the transportation perspective, but few people recommend spending a lot of time there.
You can explore train schedules on the Trenitalia website (https://www.trenitalia.com/en.html). Quite a lot of the popular small towns in Tuscany either have no rail service or are served by train stations a bus ride or taxi trip away. Hill towns, naturally enough, don't have centrally located train stations. When buses are involved, transportation is slower. For this reason, many people prefer to rent a car for part of their time in Tuscany. I haven't done that, so I can offer no advice.