How fast can I make my allergies go away?

Nasal antihistamines. Seem to have the fastest onset of action.
Goodbye allergies. It depends what you mean by going away. If you mean relieving symptoms then many of the available antihistamines can relieve some or all of the symptoms within 1-2 hours. If you are talking about curing allergies that is a different ballgame- that could utilize avoidance of allergens, allergy meds and/or immunotherapy. Even with these modalities, many patients are still not cured.