Is it normal for my eyes to get red and itchy every time the weather changes?

No. Sounds like you have an eye allergy. Best to see your ophthalmologist for a diagnosis since allergies can be see with changes in wind and humidity. Dry eye can also make the symptoms of eye allergy worse. Try some artificial tears to see if this makes a difference.
Not really. Itchy eyes are usually related to allergies. As the seasons change and new pollens fill the air, you can suffer from allergies - typically allergic rhinitis (runny nose) that can be associated with eye allergies. If the redness and itching occur with daily changes of the weather, it may be more of a dry eye condition. An eye doctor can evaluate your eyes and start you on the best regimen for you.