How does hair grow and is it similar to how grass grows?
In hair and nail growth new cells are added to the bottom. These cells are produced at the follicle. There are three stages called:


Anagen : active growth
Catagen : end of active growth
Telogen : total stop of follicular growth


I don't know what you mean by like grass but both nails and hair have a first in first out order. 

Feathers present a contrasting case: the outermost portion is the most recent. Feather growth is relatively more complex. This article is a nice read on feathers.