Nature's Way ability clarification
Does Nature's Way deal damage like an instant?

Sort of. Your thinking is correct that Nature's Way is similar to an instant in that it causes damage. If you choose a creature with 3 power as the first target, it would behave similar to a Lightning Bolt. 

On the other hand, there are many differences between Nature's Way and Lightning Bolt. Nature's Way is not the source of damage. Your creature is. This matters if, for example, your creature has deathtouch. Also, Nature's Way is a sorcery, which restricts when you are allowed to cast it.

Can my creature attack afterwards?

Sure. Your creatures don't give up and die (or tap) for no reason. Nature's Way makes your creature deal damage, but it doesn't make the other creature deal damage back. If the card said "fight" then it would become a two-way exchange.