In this picture it shows that the rate of current is the same at all points in this circuit. I am confused as to how this is so. If the light bulb provides light through resistance wouldn't this make the rate of current after the bulb (3) be slower?
No. Current is not compressible. The flow is as rigid as a loop of pipe full of water. You can't push some in one end without the same amount coming out the other end.
One of the most fundamental rules in DC electronics is that the current is the same, in a series circuit, no matter where you measure it. #12 used the example of the water pipe. That is a good one.
The current I = V/R = Bat/R_Bulb Amps, and all the current of 4 points will be the same, because the R_bulb limit the current, different R_Bulb then you will get different current, if the R_Bulb more less then the current will be more higher, if the R_Bulb more higher then the current will be more less, so in only one loop circuit, the current will be the same.
In a circuit that simple, there are only two places to measure the voltage, and the results of your measurement will be all of it or none of it. However, you can cut the wire at any place in the circuit and measure the current. It will be the same for every measurement.