I will add one thought to the already good advice. Although I am still not very good at climbing hills I did start improving after I read an article from a riding coach. The most important item in that article was about climbing and pain. The coach basically said that you will not improve much as a climber until you accept the fact that climbing a hill will HURT.
He also said that if you climb a hill and it does not hurt, then go back down and try it again, because you did not do it right. Accept this and you will start climbing better right away.