I am allergic to pine tree pollen and grass. If I stay away from areas where pines are when the pollen blows (yellow pine especially) the allergy is better.
Grass is everywhere, even in the desert, so there isn't really any change there. I avoid campgrounds that cut their grass often or at least while I am there. It is better on the road than at home because I don't have neighbors cutting their 1/4 acre plots every weekend.
With ragweed, you will have less of a problem in the west than the east. However, one might pick up new allergies as one is exposed to a greater number of irritants. I find it easier to deal with allergies on the road because I am willing to move if it becomes an issue. At home, there isn't much one can do except not go anywhere outside when it is allergy season.