Some of the puzzle solutions didn't come to me right away in the first world.  I really don't want to look at a FAQ, but I'm curious if new abilities will be unlocked later on in the game that will allow me to reach puzzle pieces in earlier worlds.

Nope, you can solve all puzzles in your first run if you can figure it out (as far as I can tell, I'm on 5 now).

There are two puzzle pieces you can't get until you have gotten another puzzle piece in the next level.

You will have to backtrack within a world, but never from one world to another.

Thanks for the help, I completed this game this weekend and it really helped to focus knowing that I wasn't missing any tools to solve any of the puzzles.

Well I cant save, so Im currently trying to beat this on one sitting. I don't understand how its possible to get some of those.

That's Braid for you. The puzzles seem impossible until something in your head clicks, and the solution becomes so fucking obvious.

Yeah, I forgot that you had to have some of the puzzle together in order to get the others in that world. That was a good mechanic, I was surprised they didn't use it more. 

"The two that  I cheated on (and now regret - DON'T CHEAT) I would've figured out if I had just taken a break or worked at a little bit longer."

"There are two puzzle pieces you can't get until you have gotten another puzzle piece in the next level."

there was only one time where i had to backtrack.