The Shrine of Resurrection
Watch the brief cutscene and you will wake up in a mysterious place, seemingly with no memory of how you got there. Once you have control of Link, play around with the camera using the right analog stick and get a feel for your surroundings. Note the HUD and your three hearts of health.
Ahead and to your left is a strange-looking pedestal. Inspect it and you will hear the same voice from earlier telling you that you need to take the Sheikah Slate. Link will automatically take it from the pedestal and put it on his hip. With the Sheikah Slate in hand, the door in front of you will open. You will enter a short hallway.
Open the two treasure chests in front of you to obtain both the Well-Worn Trousers and the Old Shirt. It's not shining armor but it will do for now (unless you like seeing Link in nothing but his boxers). Press [+] on your right Joy-Con to equip your newly salvaged clothing.
There will be a boulder in your way. Walk up to it and start climbing by pressing [X]. Then hold the left analog stick up to keep moving. Climbing will deplete your stamina, but if you stop moving, your stamina wheel will freeze. So take your time. Once you've made your way up the boulder, examine the pedestal near the next door.
At the end of this dramatic cutscene, you will see a strange old man walking down a path. This is your cue to follow, so make your way to the right and find the well-worn path (or not, since this is an open-world game after all. You can do what you want).
Be sure to pay attention to all the gatherable goodies around you, such as Hylian Mushrooms and Apples. You will want to make a habit of gathering what you can from your surroundings, as it will help you survive. All of this food can be used to heal, or can be cooked to create a myraid of other effects, so don't miss out on this easy opportunity to grab some grub.
Once you get closer to the old man's hideout, you will hear the mysterious voice you heard at the game's opening tell you to use your Sheikah Slate to pinpoint your next objective. At this point you will finally be assigned your first main quest: Follow the Sheikah Slate.