I would like to get into using motors and servos in projects. The problem is that I know nothing about the hardware. I've read a bit about the coding, I shouldn't have issues there, but I don't know what kind of motors and servos to buy, where to get them, how to calculate gear ratios, where I would even get gears, how I would hold a gear assembly all together, etc.

I was wondering if anyone out there has recommendations for books, websites, or other resources that could give someone like me a fighting chance at starting up.

This page covers gears fairly well (apart from the lack of SI units)

Lookup "torque" in wikipedia - you need to understand this.

DC motors are well described by two equations:

T = kI (torque is proportional to current)
V = kw (voltage is proportional to angular velocity)

Note the constant k is the same, its loosely called the "motor constant", and is
measured in Nm/A or equivalently in Vs/rad.

Working in SI units makes all mechanics equations nice and simple - just remember
work in metres, newtons, seconds, radians and you'll never need any conversion factors
within your calculations