initial board selection advice

Hello!
I am making a display for a computer vision/learning application.
I need to lift 15 1 pound steel balls that hang from threads
into electromagnets which hold them in place.
Then I need to release the balls simultaneously and in varying patterns.
What would be the best board for controlling one or two 250ish inch pound motors,
several position sensors, and 15 electromagnets?
motor/motors:
Length(in) 7.9
PeakTorque(lb-in) 252
BipolarVoltage(V) 10.5
Bipolar Current(A) 4.25
electromagnets:
Voltage 12
Current DC
Duty Continuous
Amps 0.08
Once I buy the right board I can practice programming it.
Thank you very much!
Chris

The Arduino does not care about the motor characteristics. The motor driver board will, and needs to be chosen accordingly.

You need an Arduino that has 15 output pins for the transistors that control the electromagnets, 4 for the speed and direction of the motors (two of which need to be PWM capable), and several for the mysterious position sensors. According to Excel, that's a total of #ERROR pins.

Any board that has that number of pins, or more, will be fine.

Thank you!

And allow me to apologize for the lack of specifics and general vagueness.
I am only a machinist and I am in over my head a little.

Do remember you can not drive those motors directly from the Arduino. You need the proper driver hardware for it, that in turn can be controlled by the Arduino. MOSFETs are great for this kind of applications.