How to identify component of velocity of a star from its red shift?
The first part of this question is already answered at How Do we know about redshift?

Regarding the second part, measuring tangential velocity is much harder, and indeed impossible for stars that are far away. Basically the method is to observe the position of the star very accurately compared to very distant stars which are assumed not to be moving and then observe it again a few years later. After adjusting for known effects like aberration and parallax, what remains is "proper motion". 

The most sensitive current telescope for this kind of measurement is the Gaia mission, which is able to measure the movemnt of stars up to a few thousand light years away with reasonable accuracy.