After killing the strange turtle-like creature on the first level, you should come to a screen with two carnivorous fish bouncing up and down in the water, along with four banks. To get to the secret room don't jump over the first well. Instead jump inside it and fall down into the secret room!
Later on the first level, look out for a ladder near some water which leads up to a platform containing an egg. Climb the ladder and take the egg, then go to the left into the water to find the submarine room.
There is also a way to find some hidden extra food. Go to the third screen of level three and jump forward hitting empty space. The hidden food should now appear.