The Doppler Shift
If an object is moving toward us or away from us, its light
will appear shifted in wavelength:

Moving towards us: blueshift

Moving away: redshift
(Think of the analogy of the train whistle.)
This doppler shift
is given by
where c is the speed of light. If the motion is nonrelativistic,
ie v_{r}<<c, this boils
down simply to
Things to notice:

The velocity in question is line
of sight radial velocity.

As defined, if v_{r} is positive the object is moving
away.
The Motions of Stars
Remember, Doppler shift only gives us a star's radial velocity.
There is also a tangential component to the star's motion.
How do we determine this tangential component? Watch a
star for a long time (years). Then measure how far it moves in the sky.
The rate at which it moves is called its proper
motion. For example, Vega's proper motion is 0.345"/yr.
Then remember that proper motion (mu) is an angular motion
across the sky. It is related to the true tangential velocity by
More things to note:

Proper motion must be converted into radians/time
for that equation to work.

We can't solve for the tangential motion until we know the
distance to the star.

At a given velocity, distant stars have smaller proper motions.
Once we have a radial velocity, a proper motion, and a distance,
we can solve for the true space motion
of the star:
Most stars near the sun have space motions of 30 km/s or
less.