After you meet the old man, a voice asks you to go to the destination marked on your map.
Get Some Equipment
Before you go to the destination marked on your Sheikah Slate map, you might want to get some better equipment. If you explore the ruined buildings in front of the temple, you can find a treasure chest that contains Hylian Trousers, which provide more defense than the Well-Worn Trousers that you found in the Shrine of Resurrection.
There are also Bokoblins around the temple that you can kill, and you can take their weapons afterward to use as your own. You can also use the directional pad to switch your weapons and shield. As mentioned earlier, weapons, shields, and bows will eventually break, so be sure to pick up new equipment often.
There are some old robot statue things around the temple. Check on them to find ancient screws and springs. These will be useful later.
You can go into the temple itself, then go out through the southeast wall to find a ladder onto the roof. Go to the right and go into the small room at the end to find a chest that contains a Soldier's Bow.
In general, if you find a Bokoblin camp, you should try to kill the Bokoblins if you think that you can survive, because they are often hiding a treasure chest that contains good equipment.
Go to the Destination
When you are done exploring, go to the destination marked on your map. Check on the pedestal, and a cut scene will occur. Afterward, go toward the hole in the floor of the tower, and press A to climb down when it prompts you to do so. Make your way down the platforms, stopping on each platform to let your stamina wheel recharge. Be careful not to fall off. Use the R control stick to look where you're going.
The old man meets you at the base of the tower. Tell him you heard a voice atop the tower. Say No when he asks if you recognized the voice. When he asks if you intend to make your way to the castle, say "I do". He says that you are high up on a plateau and you can't safely get down without a paraglider. This starts the "The Isolated Plateau" quest. Follow the old man when he starts walking, then talk to him. He will point out a glowing building.
Go toward the glowing building. You can find a Bokoblin camp along the way. If you kill all of the Bokoblins, you can open the chest, which contains a Traveler's Sword.
There is also a cook fire at the Bokoblin camp. You can cook raw ingredients here. Cooked food usually restores more hearts than raw food. To cook, go into your inventory, select an ingredient, and choose Hold. If you are already holding an ingredient, you can choose another ingredient to hold it. You can hold up to five ingredients. Then when you are next to the cook fire, press A to cook the ingredients. There are many different recipes, so try many combinations.
Then go over to the glowing building. Check on the pedestal to open the door, then step on the glowing blue spot inside and press A to enter the Oman Au Shrine.