What is the best way for me to build up arm strength to become a better climber?

You may find this link helpful as it gives many good ideas. Dont forget leg strength and flexibily are also very important.
'hang tight'

Wrists Exercises = wrist extention/ close grip bench pres. Just getting experience with climbing with help with arm strength. Also you need the strength from your back to climb and hang on, this is mainly your Lats, do lat pulldown's, seated row, one arm rows ect.
Hope this helps