I have a four year old daughter and I would like to know how to change the spelling of her name as I never really got the chance to correct it when she was born. But I haven't mentioned that we are now living in another state that she was born? Do I need to contact that state or the state I'm currently residing in? And How I go about it?

I believe you would correct the spelling in the state of birth; and change her name in her state of residence, which is your state of residence.

If you are just doing a legal "name change" you can do that in the state you are living. If you wish to correct her original birth certificate, that will need to be done in the state of birth.