Jump onto the platform to trigger the ascending spikes and quickly jump back to the right. Then, while the spikes are still coming down again, jump onto the platform and jump left off of it when it gets all the way to the left. You should miss the descending spikes if you've done it right. From there, you can either go left to where a boss is or climb the rope attached to the descending spikes. The boss is a cat that deflects all of your attacks, so as near as I can tell, you can't beat it. If you climb up, there's a sort of cloud thing you can go onto. Be careful at the bottom of the first hill, because there are spikes that will jab up and kill you right at the base.
That's part one.
PS: In case you're curious, it's pointless to kill the beaver thing to the right of the first screen. If you defeat him and continue right, you'll just fall to your death in a giant pit.