How can I wash my white socks and undershirts so that they actually
come out really white instead of just smelling clean but still looking dirty?

If the socks and undershirts are cotton, chlorine bleach can work wonders.

Wash them at the highest temperature, in a pot on the stove, if your
washing machine does not have a setting over 150° F.

Try Napisan or Napisan Plus.

I use bleach, maybe I'm just not using it right.

The pot on the stove trick might be good, I know my washer doesn't get that hot. 

Naspian Plus is expensive, seeing as my research tells me I have to
get it imported from Australia.

If there is any synthetic material in them, bleach will often yellow
them, instead of whiten them.

Kriswrite

Napisan is just stuff for soaking and washing nappies so you may find
a comparable product in your local supermarket where you live.

Nappies are the same as Diapers.

Real Simple - White Laundry Tips
http://www.realsimple.com/realsimple/content/clean/0,21770,681273,00.html

Suggest you fill the washer with soap and s lesser amount of bleach
and soak the clothes over night. Better quality cotton will be whiter
and stay whiter than lesser quality. It is kind of like buying inkjet
paper, you can get the Bright White or you can get the everyday. The
bright white is created brighter and will always be brighter. Same
goes with cotton underwear.