I am extending my floors from my kitchen into the adjoining rooms through a doorway. My question is in keeping things square. I will remove some pieces at the doorway to smooth the transition into the new room. My question is how do I keep things straight and square? Do I need to snap a chalk line or will the new pieces that I am putting in at the doorway start my straight line and continue off it?

It's a matter of chalking lines at strategic walls and points. Your outside walls are your straightest walls.

It depends on the direction the exiting flooring is running. If the new area needs to be weaved into the exiting then you need to snap a chalk line even with the first board you put in. If the flooring is running against the doorway then just continue it off of the existing. I hope this makes sense.