You begin in the Sanctuary with the Priest. Zelda will then come in and talk to both of you, and then latch onto you. She will follow you for the rest of the Week, but you aren't restricted like you are with other following characters, so you can go into caves etc. as if she wasn't there.
The moment you go outside, your character sprite will have an incorrect palette. There's nothing that can be done about this at present, and unfortunately you're stuck looking like this for the rest of Week 4.
If you look at your map you'll see that in this final Week you'll be able to explore all of Hyrule, except for part of Death Mountain.
First off go to the left, up, and right and ram into the rock-pile for a Great Fairy and, you guessed it, a Piece of Heart. Now go left and up and into the cave behind the Lumberjacks' house for another Piece of Heart.
Now go back to the graveyard and push open the top-left grave. Go up through the door and up the stairs and then over to the side. Get the 100 rupees and Bottle from the chests, then go down and exit. Fall off the cliff and make your way to the top-right grave. Ram it open and go inside.