You're heading to VAST FOREST to find BERLY'S BALL, aren't you? Better be careful! There are enemies afoot! If you're not sure how to defend yourself, you should read this "ALL ABOUT BATTLES" book I found! It's right here next to me if you need it.
VAN and BERLY are playing tetherball when VAN punts the ball away. Help BERLY retrieve the ball. BERLY will temporarily join your party.
Afterwards, DAISY, who was previously blocking the path, has stepped aside from the west exit of the PLAYGROUND. Enter the VAST FOREST from here and you will encounter a LOST SPROUT MOLE. You cannot run away from this battle.
Once you complete your battle, keep going west until you see a path going south. Enter here and you will meet GRANDMA outside of BASIL'S HOUSE. After speaking with her, you will complete the quest. You will need to return to the PLAYGROUND to drop off BERLY.
Nice, you got BERLY's ball back! She's so ecstatic, she even wants to give you something as a thank-you. How rare for her!