Instructions
Watch for Professor Pester starting at approximately level 15. You will have sour pinata invasions before this level, but Professor Pester himself will not bother you until then. Pester appearances are random, so you may not see him until a much higher level.
Purchase decoy pinatas from Costalot's store in the Village early on to stop Professor Pester from killing rare and hard-to-get pinatas. You may need to have multiple decoys down at the same time, as ruffians often venture into your garden before Pester appears.
Buy a mine from Willy Builder and place it. The mine is available once you hit level 26. Go to Artfur's Inn and hire at least one diggerling. Once the diggerling finds a Dragonache egg, go to Miss Petula's Paper Pets and buy a Cluckles. Send the bird to the egg to hatch it. After the Dragonache matures, he will scare Pester out of your garden, often setting him on fire.
Go to Costalot's store in the Village if you want to eliminate Professor Pester for good. Buy a Captain's Cutlass and place it in your garden to prevent him from entering. You don't get access to Capain's Cutlass until level 30.