I played the demo, beat world 2, saw the dino, beat the single demo level of world 3 and then purchased the game. I instantly got 2 achievements and access to worlds 3 + 4. Is this how the level access plays out in the game if you start from scratch? Also, where is world 1?

You start in World 2. When you get to the end, it unlocks World 3. Finish World 3 and it unlocks World 4. Finish World 4 and it unlocks Worlds 5 and 6. World 1 gets explained later in the game, but I don't want to spoil anything for you.

So yes you actually started from the beginning of the game in the demo. Congrats on the purchase. You won't regret it.

Thanks for the answer, but is it normal to get world 4 after just 1 'room' of world 3?

I don't know. I bought the game outright and then started playing, so when I entered World 3 I didn't exit it until I finished it and then World 4 was available. I suppose it would have been the same for me had I quit mid way through World 3 to get back to the house. I do know that World 5 and 6 open up simultaneously and can be played in either order.

I think you can get to the different worlds w/o doing much in the previous worlds.... Johnathon Blow wanted to make sure that if you got stumped on a puzzle, you could go further into the game w/o being forced to solve specific puzzles at a given time.