Cheap Apartments For Rent Near Taunton Ma . Payment (low to high) 22 lawrence st apt 3r, taunton, ma 02780. See all 43 apartments under $1,000 in taunton, ma currently available for rent. Pine Grove Taunton, MA Apartment Finder from www.apartmentfinder.com Taunton single family homes for sale; Each apartments.com listing has verified availability, rental rates, photos, floor plans and more. An apartment unit for rent in this area is on average $2,398.