Is it safe to connect the three cables on the right side of the resistors to the same GND on my RPi, or should I put them on separate ones? (I know very little about electronics, but I'm trying to learn.) Also how do I know how many components I can safely use on my breadboard to not damage the RPi, if there's a limit?
All the ground pins are connected. It doesn't matter which one you use.
Use the power and ground rails on the left and right of your solderless breadboard. Connect the blue rail to one of the Raspberry's GND pins.