Unfortunately, the majority of museums do not allow cameras or photos inside. You can't take pictures inside the Uffizi museum although you CAN take pictures in the corridors toward the outdoors, that is to the view, for example of the Ponte Vecchio; the Accademia Gallery (all pictures of the David were taken "illegally", there is always someone patrolling and people do take pics but that first one has to be good because then they practically yell at you); for some reason within the SMN church you also can't take pictures. You cannot take pics inside the San Lorenzo church or the Medici chapels. No pics in the Pitti museums or in the Orsanmichele church.
You can take pictures as long as without flash inside the cathedral, inside the Santa Croce church and inside the cells of San Marco's monastery. You can take pics in the courtyard of the Bargello museum but not the museum's rooms. Haven't been in the baptistery in a while, can't remember that one..... the Museo dell'Opera is another museum where you can't take photos. Ah, you can take pictures inside Palazzo Vecchio, no flash of course but the rooms are pretty bright. You can take pics of the frescoes in the Brancacci chapel without flash which is really nice but light is not very bright in the side chapel, when I've tried before the pictures came out all blurry.
Ok if you have a specific museum or church in mind, do ask since I might have forgotten a major one right now!