to prove the Ohm's Law
:( Any solution using the Arduino will run you in circles...
With the Arduino, you measure current by measuring voltage across a known resistance and applying Ohm's Law. I don't now how Mr. Georg Ohm did it, but I assume you can look that up.
I'm not sure this would qualify as scientific proof, but you could get a [u]mechanical ammeter[/u] to measure current and demonstrate Ohm's Law. A mechanical meter works because meter deflection is proportional to magnetic field and the the magnetic field is proportional to current.
There is a voltage drop across the mechanical meter but the current reading is not "derived" from the voltage. In fact, a mechanical meter works by deriving the voltage from the measured current! (The opposite of a digital voltmeter.)
1.How do I measure the Current Flow (I) of the circuit?
You temporarily "break the circuit" and insert the ammeter (or multimeter) in series with the existing components. Multimeters don't have a common ground with the circuit being measured, and that's important (most are battery operated). Digital multimeters measure the voltage across a known (small) resistor and they apply Ohm's Law internally to calculate and display the current.
Or, you measure voltage across the existing resistor and apply Ohm's Law. :D
2.How can I change the Current Flow of the circuit? (How to change the current flow rapidly)
You can change the voltage rapidly by using a digital-to-analog converter, or an AC voltage is "rapidly changing". Or, you could use a digital potentiometer to change the resistance rapidly. Or, you could build a digitally-controlled current-source (a bit tricky but not impossible).
3.How to get the output in this form.. (V = 2 and I = 1) "2/1 = 2"
I don't understand your question, but microcontrollers (including the Arduino) are really good at math. You can send the output to your computer via the serial monitor and display whatever you want, or you can add an LCD to your Arduino circuit.