There are a lot of free activities in Rome but you have to have the right info to take advantage of them. For instance, you can save on Rome’s most expensive museum ticket, the Vatican Museums’, if you visit them the last Sunday of the month. Palazzo delle Esposizioni has free entrance for visitors under the age of 30 the first Wednesday of the month. But most of the possibilities are offered by private galleries: they often host small exhibitions that are free because they also are in the business of selling art works. FYI, Gagosian is also in Rome.
Another occasion is represented by auction houses who show the pieces on sale for free a few days before the actual auction. Christie’s, Sotheby’s and Bloomsbury all have a site in Rome.