You can unlock the various gold tools by completing the corresponding requirements, as indicated below.
Shovel - Purchase 50 fertilizer packs from T&T Emporium.
Fishing Rod - Catch every type of fish and talk to Chip at the next fishing tournament.
Net - Catch every type of bug, then talk to Nate at the next Bug-Off.
Watering Can - Receive and maintain a perfect town rating for 15 days in a row, then talk to Isabelle about the town rating at the Town Hall.
Axe - Purchase 50 tree saplings from the Gardening Shop.
Slingshot - Shoot down 16 balloons and it will appear in the air, suspended by three gold balloons.