Carrara is the obvious first stop — just 6 km away, the town and its surrounding quarries are among the most remarkable industrial landscapes in Europe. You can visit the historic Fantiscritti quarry basin, see marble being cut at one of the many workshops, and walk through a centre whose streets are dusted white year-round. The cathedral, built partly from locally quarried marble, is worth a look.
Lucca makes an excellent half-day or full-day trip: the walled Renaissance city is about 45 km south and one of Tuscany's most rewarding smaller cities, with a perfectly preserved oval piazza (the old Roman amphitheatre) and a cycle path running along the tops of the intact city walls.
The Cinque Terre villages lie roughly 40 km north along the coast. From Massa it is easy to reach La Spezia by train and then hop the local service to Riomaggiore or Vernazza; going early in the day avoids the worst of the summer crowds on the clifftop paths.
The Apuan Alps Nature Park starts almost at Massa's doorstep. The park contains dramatic marble landscapes, deep gorges, and the highest peaks in Tuscany outside the Apennines proper. The road up to the Passo del Vestito offers sweeping views over both the mountains and the Ligurian Sea.