You can't "delete" save data like a file—Nintendogs + Cats stores saves directly on the game cartridge. Here's how to start fresh:
Result: Your old save is permanently replaced. All pets, items, money, and progress are gone.
Some versions allow transferring pets to a new save via local wireless or StreetPass before resetting. Check the in-game menu for "Transfer" or "Move" options if you want to keep specific pets.
Warning: This is irreversible. Only start a new game if you're certain you want to lose all progress.