new project idea... need hardware advice

I am very new to using computer boards. But have a project idea and was wondering if someone would help with a fairly cheap solution. I took apart an HP printer and acquired a few motors, and sensors. Now I want to set them up to follow a set of commands. The motors are all 12vdc 2wire motors. And the sensors are rotory and linear plastic positioning... Will include pictures, and model numbers soon. What I want them to do is...
1 move linear drive to linear sensor stop point 1 using motor 1
2 move linear drive to stop point 2 using motor 2
3 rotate rotory disk to next of 4 stop points using motor 3
Want to know what board and shields will be needed to start this project... Will probably ask for install and programing advice later.
Final goal read 2 sensors with 3 wires each and drive 3 2 wire 12vdc motors... Which Arduino parts will I need?

Linear sensor
HP Tyco C9017-80014

Rotory sensor
HP Tyco C9017-80055

Motors are both HP belt drive 12v and will be adding 1 more for function 2 soon.
Pictures are very large... Sorry just uploaded them to a project site from my phone

I took a small Canon printer to pieces a few days ago. It sounds as if it worked similarly to yours. The position control was done using (for one motor) a transparent plastic disc about 65mm diameter with very large number of fine lines printed around the edge to interrupt the optical detector. There was also a length of linear plastic strip with similar markings for positioning the print head horizontally.

I suspect there is some very sophisticated software to control the motors using those encoders. Saving 10p by using a cheaper motor will pay for a lot of software development when you sell hundreds of thousands of printers.

Some Arduino code with a simple interrupt routine can count pulses from the optical detector.

I will leave it to others to advise you how to turn that information into useful motor control. I have no idea, for example, if it is actually possible to stop the motor at a specific location or if the system relies on doing things as the print-head (for example) passes by. Like the paper-boys you see throwing papers as they cycle past houses in American movies.

...R