A high quality vacuum cleaner is actually your best defense against hair, dust, and dirt in your home.
Because of your allergy sensitivities I would suggest a central vacuum system , if possible, or a Miele brand vacuum cleaner. Miele's have superior filtration and capture systems that are sealed, preventing dust from leaking back into the air you are breathing.
Follow your vacuuming with a traditional wet mop. Stay away from dry mop products as the chemicals in them can irritate your allergies, as well as leave behind residues that are highly toxic to pets and small children.
You may also want to look into using quality filters in your heating and air conditioning system, as well as room air purifiers, such as the Honeywell 50250-S True HEPA Air Purifier, to reduce irritants throughout the house and improve indoor air quality.