I'll add my tuppence worth:
Assuming standard bi-polar motors: NEMA stepper motors have two important descriptions you will need to review for your application. 1. The physical size of the mounting face (NEMA 8, 11,17,23,34) all refer to the dimension of the front face. 2. The Force/current rating, which will in part determine the number of coil stacks used in the windings and so defines the height of the motor. (lower the impedance, higher the force, higher current requirement, more windings = bigger motor). Most of the standard motors are designed to 1.8 deg full step, but driven by a micro-stepping unit you can improve the resolution further.
I have chosen NEMA-8 for a telescope focuser, with 0.25Nm with roughly a 50:1 gearing ratio . This tiny (6cm) motor is perfect for the job, but still well over powered/engineered. A stepper motor from a CD player would have just about been enough.
I would expect that NEMA 17 would be about the size (maybe even smaller would do) for your application (assuming you are using worm drive gears), but you might need a higher force depending on your mount loads you are planning.
My 3d printer uses NEMA 17, but others work well on NEMA11.
I'm running NEMA 23, 8Nm for a CNC machine, it has over 300kg of force on the spindle to drive the axes.
You might also consider motors with built-in position decoders - they do exist but only for the large motors - a bit pricey too, and interfacing them to a raspberry pi might prove interesting. This would be the approach I would investigate if I were considering this project.
One final point. I would strongly recommend using a proper stepper motor micro-controller units which is able to drive higher voltages (and I would recommend using 24v - all motors will tolerate this). These controllers avoid burning the steppers out by regulating the current flow allowed into the motor, and so get far more force/power for the motor size (important if size is an issue for you). These controllers also lower the power consumption with idling and current profiling on load, which keeps them cool (a point that wbirk has made), and a good quality one will also opto-isolate the inputs/outputs to protect your equipment from motor back EMF.
I've not used the DRV8825 boards myself, they are going to be much better than and Easy Driver boards, and will profile currents, but I'd be concerned about the heat/size. My 3d printer uses DRV8825, and they have been known to fail under loads for long periods of time. I have sourced drivers off ebay for £5 that are the size of 4 flat stacked of iphone 6s, and seem to me to be more robust and suitable for passive cooling. I would only use DRV8825 with heatsinks stuck in the IC and a small fan running over them.