Don't forget you have an 8-way joystick. Make use of it. If an enemy is at any 45 degree angle and in your way, shoot it! Use this when trying to get objects in the spider's web. Often the spider will attack at an angle.
Many people don't realize that Peter can swim. After all, you are a rat and a rats can swim.
When you are in the water you can stun and ride the alligator. If you do manage to agitate the alligator and he starts chomping at you in the air, don't worry about it. He can never get you that way. Only when you are in the water or drop directly into his mouth.
Gravity can be your friend when dropped from a flying creature. If you get knocked loose or if your flying buddy is about to let you go, THROW AN OBJECT DOWNWARD IMMEDIATELY. There is a very good chance you will hit another attacking flying creature and now, you dropped on it and it's carrying you! This is particularly effective with owls in the tree scene.
Don't let the slick surfaces in the sewer intimidate you. You can still jump left and right when on top of them. This is advantageous when trying to avoid the alligator, the spider and other enemies that may be right on your tail.
When jumping left, right or falling, pushing up on the joystick when Peter is right over a ladder will make him grab onto it. This is useful in avoiding enemies that are right below you and off the screen.
In higher levels (known as 'waves' in this game), Peter gets stronger and can jump farther and higher. Use this new "super jump" ability to jump straight up to levels above him, to quickly avoid the attacking creatures and move farther left and right across the screen. This ability is extremely advantageous in the tree scene. Peter can jump clear across from one side of the tree to the other.