Stockinette Stitch
Reference for the below knit kits:
 
There are two methods for weaving in your ends for stockinette stitch below.  Method 1 is recommended for beginners, but Method 2 is worth a try as it hides the ends really well.

Method 1: Weave in your Ends on the Vertical
If you stretch out your work like the first image on the left, you will see bars between each knit stitch.

Thread your end through the embroidery needle (a contrast yarn is used in the example below to make it easier to follow along).  Picking the nearest column of V stitches, bring your needle through every other bar about five times.
   

Bringing your needle perpendicular, thread it under the nearest adjacent upside down V.  Going down, thread it through ever other bar in the adjacent row of V's.
   

Go up and down one more time and then cut the remaining yarn.

Method 2: Weave in Ends on the Diagonal
The "wrong side" of stockinette stitch (i.e. if you flip a hat inside out) looks like the below image with alternating bumps that resemble frowns and smiles.


Thread the embroidery needle with your end (if your end is on the right side of the fabric, bring it through to the wrong side).

Bring the needle through the frowns going diagonally.  This means that you will bring the needle through the frown of a row.  Then, depending on what direction you're going, bring it through the frown in the row above or below. 

This next frown should not be directly above the first frown you came through, it will be either to the right or left.


Once you have threaded the needle through at least four frowns, pull your end all the way through.


Then, thread the needle going the opposite direction that you just came from - either down or up - through the smiles directly adjacent to the frowns you just came through.


Once you have threaded the needle through at least four smiles, pull your end all the way through.  Trim your yarn leaving about 1/4 inch.