Hi!
Here are a few suggestions.
-make sure you're bike is fitted properly - seat height, etc. Christine has a pretty good video
here.
-you may want to raise your handlebars up a bit so that you're not leaning forward too much
-keep your grip loose and elbows bent
-when you stand up keep your center of gravity back and over your pedals and don't lean and put weight on your handlebars
I have a bit of carpel tunnel in one wrist from being on the computer constantly so I try to do these things as well. One additional thing I do is to double up a towel on the handlebars for extra padding.
Hope that helps!