Stepper, servo, or continuous motion??

I've been watching a bunch of videos, but it seems that everybody has their own custom version of the project I have in mind- an IR controlled windowshade.

From what I can tell, it seems like I should use a continuous motion servo- not as limited as a servo, but not as complex as a stepper.

Ideally, I'd like to create a battery-powered (maybe even solar!) contraption that can use my existing TV remote. Doesn't seem like I'd need something as large as an Uno, but the Nano might not be sufficient.....still trying to decide! 

Think I also need a driver, again not sure what kind, this is all very new to me, though I've been watching the tutorials. Don't know if everything I need would be in a starter kit.

Thanks for any an all suggestions!