An option from Accy, especially with a bus pass, is the X41 down the M66 to Prestwich, then a short walk to catch a north-bound 135 which stops on Manchester Road close to Gigg Lane. Traveline at
Welcome to Traveline North West will give bus times. Gigg Lane's postcode is BL9 9HR.
As Wynonie says, there are some good pubs on the way to the ground but two good ones in Bury itself are the Trackside on the East Lancs Railway platform at Bolton Street Station, which is on the bus route from Bolton and is accessible from steps at the end of Bank Street (BL9 0DZ), and the Automatic in Market Street (BL9 0BW). Both serve food and real ales from local breweries.
Be aware, if you're taking a car, that these days most of the street parking around Gigg Lane is for residents only.