Why go
The Sienese school includes Duccio, Simone Martini and the Lorenzetti brothers, but their works tell different stories in different places. Palazzo Pubblico makes civic painting central; the cathedral combines sculpture, marble and frescoes; the Pinacoteca concentrates panel painting when open. Pick a setting first, then follow the artists across it, rather than trying to cover every name in one pass.
The full story
For Sienese painting, begin by asking what the work was made to do. In Palazzo Pubblico, Ambrogio Lorenzetti's Good Government frescoes turn civic ideas into images for a governing space. In the cathedral complex, art is tied to worship and the building itself: the inlaid floor unfolds beneath visitors, Nicola Pisano's pulpit is sculpted for a church interior, and the Piccolomini Library surrounds them with painted scenes. Duccio and Simone Martini belong to the city's medieval painting tradition, whose gold grounds and precise figures reward close comparison. The Pinacoteca Nazionale brings many Sienese panels into one collection, though its reopening status needs checking before a visit. That museum experience differs from reading a work within the palace or church for which it was made. If time is short, choose one civic interior and one religious or gallery setting. The change of context will teach you more than a long list of artists.