I'm pretty self-conscious about my race
I'm not sure I understand. Why does your race make you feel self-conscious?
All my close friends are white, my parents are white, I went to a white school
It's totally normal to feel self conscious about stuff, especially when it's something that makes you seem different than the people you're around.  It's important to realize that there's nothing wrong with having a different race though, it's what's on the inside that matters!
I'm not sure you understand what I'm saying. It's given me a lot of self-esteem issues.
It might be hard for me to truly understand your situation, but I think building self-confidence means you should work on undoing shame that you might feel. I'm sure it will help you accept your differences.