There's really nothing to do differently. Nothing has an impact. You can do things out of order, but in BOTW (like most Zeldas) that's about as impactful as deciding whether to add peanut butter or jelly first in a PB&J sandwich.
Thinking about playing again, I'm just trying to figure out a way to deal with the game's problems. There's fun content with the shrines and DBs, but the overworld and grinding really wear things down and make the game almost...unfun. I figured I'd just pick up some DLC gear and never deal with...