My whole family has allergies/asthma. I've heard all my life that moving out west (us) cures them. Is that true?

If only. Allergies can sometimes be successfully with long-term allergy shots but can rarely be "cured." similarly, asthma can be treated and well-controlled but cannot be cured. Kids can sometimes grow out of both allergies and asthma, especially if they have it early-on.
Absolutely not. You can't be allergic to things to which you haven't been exposed. People from the midwest have no symptoms when they visit the west, but become sensitized to the pollens found in those states after about 3-4 years. By the fifth year, they're just as symptomatic as they were in the midwest, so they move back to reconnect with their old friends.