Pick up the food bowl and walk right into the house. Give the bowl to Charlie Brown, then wait and eat the food. Walk left to the hut, pick up the typewriter, use it, pick up the letter and give it to Charlie Brown.
Eat the cookies and take the jar with you to the pond which is towards the right. Use the cookie jar to catch the frog. Walk left to the first house, and walk down the path. Walk out to the back yard and leave the frog in the jar. Take the umbrella.
Walk left out of the house and back to the street, then walk all the way to the left. Use the umbrella to walk under the rain cloud and get the ball. At this point it’s best to pickup the ball, kick it, then pick up the catapult and walk towards the ball. Repeat this so you kick the ball all the way to the right while transporting the catapult. This way you can get both items to the river reasonably quickly.
When you reach the river, kick the ball into the river and pickup the coin. Jump over the barrels (jump when the object you want to land on is at the top of it’s motion, and it will have bobbed down and back up by the time you land on it).
Toss the coin into the well. Jump back left across the river (or deliberately fall into it to appear on the bank again - much quicker!) and get the catapult. Jump back right to the well and burst the balloons with the catapult. A pump will be attached to one of the balloons. Collect the pump.
Cross the river again and walk into the first house. Walk out the back of the house and pump up Lucy’s paddling pool by holding fire and waggling the joystick. Go left one screen and take the hose to the right. Fill the pool with the hose water. Get the frog in the jar and scare Lucy by letting the frog loose.
Pick up the horse and take it to the tree with the kite. Get the pump and pump up the horse. Jump on the horse to get the kite. Go to the rain cloud with the kite and give the kite to Charlie Brown. Follow him, pick up the key, and use the key in the school lockers. Get the blanket and give it to Linus. You’ve completed the game!