(Automatic Bike Project) amount of torque required to change gears?
From working on bikes, the amount of force needed to pull a gear cable is more than I can apply gripping the cable with my fingers, but is easy to apply gripping the cable with pliers. So probably about 10-20N. But almost all that force is required to oppose the return spring in the derailleur. Remove that spring and the force drops by an order of magnitude. And as Kibbee says, you can use leverage to change that force (but you still need the same amount of energy).

The easy way would be to use an existing gear cable and gear lever attached to the rear derailleur. That way you don't have to mount your motor right down next to the rear axle, where it's vulnerable to shock and dirt.

If you use a winch drum attached to a stepper motor it'll be fairly easy to make everything work. The issue with that is that the holding torque will almost certainly be greater than the stiction of the motor, so you would need power to stay in gear. Or gear down the stepper motor, or use a mechanical brake with a solenoid to release it when you want to shift. But that's possibly more complex than a student project can easily build.

A different way would be to use a common two-lever shifter with a motor to push the levers when you want to change gear. That's a low-precision task, but you will need considerable force. If you used a fairly hefty RC servo motor I think it could work, and within budget. Then drive that off a microprocessor that has a PWM output to make the servo work (there are Arduino libraries that do this OOB, it's not hard)

A much more elegant way would be to modify a derailleur directly, removing the return spring and using a motor to move it backwards and forwards. This is what the electronic shifter setups do. If you use a stepper motor and drive "off the end" to position it, you should get reasonable reliability at low cost and without having to build sensors and servo rigs to locate the chain exactly on the drive cog. I fear you might need a brake again, but I'd build the first version without it. At worst, you can do your final report saying "it works but needs a lock to hold it in position over bumps".