as said, a tour of rugged Garfagnana could be a great choice. You need a car, and this is a tour i suggest to you: start with a visit at the ponte del diavolo, in Ponte a Moriano, (approx 20 kms from Lucca), than a visit at the unknown Eremo di Calomini, built against a store, than have lunch in Barga, lovely medieval village with one of the most beautiful romanic stile church i have ever seen. On the way back down to Lucca, visit Bagni di Lucca, a thermal village and the chain bridge in Fornoli. This with regard to the day trip outside the town.
In Lucca, you have a plenty of things to visit: the walls, its churchs, its squares and towers, its museums, the botanic Garden (not a big one, but a very pleasant and peaceful place), or just lose yourself in the narrow streets of this fantastic town, and you will see that it will deserve to you many surprises
This is wonderful! Thanks!
As you say that you are staying in an apartment, I'm assuming that you might have some meals in, if so this info might be of some use; here is a list of some places that I frequently shop at when staying in Lucca each summer. Tuccori on Via Santa Croce for prepared food, deli items eg. grilled veg, salads and cooked meats, ragu (I take loads back home) and if you like lasagne theirs is the best both my husband and I have tasted, usually they make it on Thursdays and Saturdays. They also have a butchers counter at the back of the shop from which I found the meat to be of very good quality. Veg and fruit from Ortofrutta in Piazza Salvatore, you choose your produce, then get it weighed and are given a ticket which you take to the cashier and make the payment. For focaccia and breads Forno Casali (personally I think it sells the best in Lucca) which is near the end of Via Guinigi also Forno Amedeo Giusti on Via Santa Lucia (behind Piazza San Michele). Cakes, Pasticceria Buralli in Piazza del Carmine, Pasticceria Dianda in Via della Rosa and for lovely tarts Panificio Chifenti on Via San Paolino. For wine we find that the supermarket Esselunga, just outside the walls on Viale Carlo del Prete, has a good selection at reasonable prices.