All I can say is walk forward. Don't worry about any of the animals - they're completely harmless.
Eventually you'll reach an clearing with a totem pole on each side. At this point, the Gatekeeper enters, and Heather appears above you. The Gatekeeper's hands are his only harmful body parts, and after jumping a few times (in your direction), he'll turn towards you to fire his arm at you. A good way of dodging this is to remain close to him, and when he turns around, walk behind him so the arm cannot reach you.
Now, notice that he has that big thing on his back - this is his current weak-point, but all you can do is position the Gatekeeper so that Heather can throw one of her keys into it. To do this, see which side of the Gatekeeper Heather is on, and be on the other side. This way, when the Gatekeeper turns to fire at you, his back will be exposed to Heather. You should keep in mind that Heather doesn't wait around too long on one side before going to the other, so work out roughly when she'll switch sides so you can be on the correct side of the Gatekeeper.
Once he's beaten, he'll fall of the stage, and there'll be a quick scene in which the backdrop falls apart to reveal the Dark Demon's castle. At this point, the the claw of the Nasty Gatekeeper will grab Heather and he'll jump onto the stage and open his mask. This is when you need to act, so get under him and hit his exposed face. Whenever you do hit him, he gets very angry, so while he's blowing steam (literally), get as far away as you can from him, because he starts firing his arm repeatedly at your head. Since he is aiming at your head, you can dodge him by hopping in the air slightly (tap 'Jump' lightly) just before it fires, and the arm will fly over you. Keep hopping again and again (each time just after his arm retracts over you) until he's fired five times.
Next, he'll get ready to fire an arm at you (slower now) or may start jumping around first, while spinning his free hand. If he does jump, then get to the side of screen, away from him, and, when he's about to make the cornering jump, run towards him so you can get under and past his spinning arm. When he stops, he'll finally get ready to fire his arm at you, so make your way to the end of the screen and jump over the arm as it passes you (if his free arm is fired and leaves the screen, it'll come back with a Headcase, and vice-versa). While his hand is stuck, make your way towards him. He may fire another arm, but he'll eventually expose his face in order to drop a bomb, so attack him. But be aware - the bomb will still be falling, so as soon as he is hit, run away, making sure you jump to avoid the shock-waves from the bomb.
Again, the Nasty Gatekeeper will get angry and fire rapidly at you, and the whole sequence shown above will repeat until he is defeated. Just an important note - try not to hit Heather, otherwise the screen will explode, dealing you (and only you) damage.