Usually Chris Carter is here for the community when easing into a new game, but I guess he just doesn't have the heart to work up that spark he has so many times before, so playing through the game and having a blast, I thought I'd share some tips, which are story-spoiler free, in his place!
Don't be afraid to explore without renting all equipment! If you ever find yourself making it to a dungeon and you're ill-equipped to handle it, there will ALWAYS be a weathervane at the entrance. Simply activate it and warp back with what you'll need. However, I would recommend to always travel with the Bombs, Hammer, and Hookshot, as these are vital for opening up pathways and the like, which usually lead to finding dungeons, quests, or mad money. (Mostly mad money.)
Travel with a fairy in a bottle when you rent. This is a nice safety net to have since you don't lose the items if you're brought back by a fairy, but since it only recovers 5 hearts, you may want to take it as a warning to get out of there! (Make sure you buy a Scoot Fruit from an item shop!) You'll receive the net from the house with a giant bee on it in Kakariko Village.
DON'T FORGET TO BUY A SHIELD. [s]The game doesn't outright give you one (So far as I've seen)[/s] (It's going to be a while until you can find one!) and it took me 5 dungeons in to even realize something was missing. They're 50 rupees at an item shop, don't forget!
If you see a rock that glitters, DO NOT throw it. Slash it frantically instead and get all of the Rupees from it first. Then you can throw it!
Keep the environment in mind when you're fighting. If you have a swarm of enemies coming at you or there's one of those that just won't die, you can usually always hit them off an edge, or now, you can just escape using the walls (You can't be damaged). A trick I've been finding pretty handy is getting enemies attention, placing a bomb down and then hiding in the wall as they sit there. You've gotta be careful with the timing though, because otherwise you'll only trap and hurt yourself!
And there you go! I may have more to add to this list once since I'm still playing the game (Working on my 3rd dungeon in Lorule now), but hopefully these help a few of you who are playing as well. Feel free to share some tips of your own!
EDIT: Since the listing just shows up as text instead of formatting, I've just taken all of the BB code out. Sorry for the lack of prettiness!