The first trial involves crossing the river of sand. This part of the trial is easy enough. Just use your Longshot to aim at the posts and crates on the opposite side of the river. The river of sand acts as an equipment check – it ensures that you do not proceed to the Spirit Temple until later on in the game (namely, after you have completed the Water Temple).
The next trial involves walking from market to marker through the sand. You will come across a sign pointing to a shop which you can ignore entirely. Following the sign leads to a weird man sitting on a magic carpet above a pit of sand. You can equip your Hover Boots and run over to the magic carpet and then speak to him to find out that he will sell you a mystery item. He won’t tell you what it is and he asks for 200 Rupees to buy it. The items that he sells you are just standard items that you can pick up anywhere. Return back to the sign post if you did venture out and continue along the pathway.
Follow the flags until you reach a large stone structure surrounded by flags. Drop down the hole to find a Gold Skulltula. Destroy it and obtain the token (number 91 of 100). Light both of the torches in this room (using either the Fire Arrows or Din's Fire) to reveal a small treasure chest that contains a Purple Rupee (worth 50 Rupees). One of the pots in this room contains a Fairy which you can grab in one of your Bottles if you need one. Climb back up out of the stone structure afterwards.
The sign on the top of the structure provides a clue for what to do next for the second trial (Phantom Guide): “One with the eye of truth shall be guided to the Spirit Temple by an inviting ghost”. Use the Lens of Truth to reveal a Poe floating nearby which will encourage you to follow it as quickly as you can. Be careful of the Leever enemies (the green blobs) that appear as you follow the ghost. You can dodge them by rolling to the left or right as they approach, but be careful not to stray to far away from the Poe or you will be left behind.
Following the Poe will eventually lead you to the Desert Colossus.