Why won't my motor run?
An NPN bipolar transistor requires a current to flow from base to the emitter which is then multiplied by the beta of the transistor controlling the maximum current which flows from the collector to the emitter.  A 2n2222 beta is about 100.  Let us calculate the amount of current which will be allowed to flow from the collector to the emitter of the 2n2222 transistor.  First calculate the current from the base to the emitter:
I(be) = (5.0 volts- 0.7 volts) / 1,000 ohms
I(be) = .0043 Amps

Use this current to find the maximum current flowing from the emitter to the collector:
I(ce) = 100 x I(be) = 0.430 Amp or 430mA

We see that the transistor does not provide enough current to drive a 600mA motor.
We need to reduce the value of R1 such that I(ce) is at least 600mA which means that I(be) needs to be at least 6mA.  Rearranging the earlier equation to find R1:
I(be) = (5.0 volts- 0.7 volts) / 1,000 ohms
I(be) = (5.0 volts- 0.7 volts) / R1
R1 = (5.0 volts- 0.7 volts) / I(be)
R1 = (5.0 volts- 0.7 volts) / 0.006 Amps
R1 = 716.67 ohms

However, because of the large amount of I(ce) current necessary to drive the load, we find the transistor reaching it's limits and the beta of the transistors drop apprecably:

This diagram is a little hard to understand.  And there is no curve for 600mA.  But we can get an idea of what to expect if we look at the 500mA curve.  We see that in order to get less than a volt drop across the transistor (V(ce)) we need to pump in more than 10mA (I(be)).  To get similar results for 600mA we would likely approach 20mA and now we are dangerously close to the processor pin current driving capabilities.
A common alternative is to use a Darlington Transistor configuration:

The resulting beta is essentially the product of the individual transistor's betas.
Let's do some rough calculations.  If the 2n2222 that switches the load current has a lower beta of about 10, we would need about 60mA to switch a 600mA current.  To generate 60mA through the transistor connected to the processor, we would need about 0.6mA from the processor.  That's really small and we can afford to triple it to ensure reliable operation.  So let's pick 3mA:
R1 = (5.0 volts- (2 * 0.7 volts)) / 0.003
R1 = 1200 ohm

So a 1K ohm resistor should likely work.