After you go through Pondside Cave and further explore Great Sky Island, you find Mining Cave.
Use a Cart
Near the entrance of the cave, there are mining carts and fans. You can attach a fan to the back edge of the cart, then place the cart on the rails, and hit the fan to make it move along the rails.
Eventually, the cart will hit an obstacle. Be sure to hit the fan to stop it. There are no rails beyond the obstacle, so just leave the cart behind.
Light Up the Cave
Talk to the Mining Construct up ahead to learn about using Brightblooms to illuminate the cave. You can attach Brightbloom Seeds to your arrows, or you can just throw Brightbloom Seeds to make them light up the area.
You can hit the ore in this area to get chunks of Zonaite. Farther ahead, use the Fuse ability to attach a rock to a wooden stick, turning it into a Rock Hammer. Hit the ore with the rock hammer to get more Zonaite.
Keep an eye out for the Bubbulfrog on the ceiling in this area.
Go through the nearby door and light up the area, then talk to the Construct at the forge to learn about Zonaite. You can spend your Zonaite here to purchase Zonai Charges and Crystallized Charges. Both are useful, so feel free to purchase some now.
Talk to the Construct near the mine carts to get some Zonai Capsules that contain Fans. Zonai Capsules can be found in your inventory. If you use one, the device can't be put back into the capsule, so be sure to only use them as needed.
Use a Fan capsule and atach the fan to the back of a mine cart, then place the cart on the rails nearby. Get in the cart and hit the fan to start it.
At the top, you can talk to Rauru. Farther ahead, there is an orb-shaped building. Talk to the Maker Construct next to this building to learn how to use it. You can drop Construct Horns and Zonai Charges into the cookpot-shaped bin to convert them into Zonai Capsules. If you only put in one Zonai Charge, you only get one Zonai Capsule, but if you put five Zonai Charges in, you get more than five capsules. This dispenser can dispense Flame Emitters, Fans, and Portable Pots. You can look at the map to see the full list of items that the dispenser dispenses.
Help the Korok
Nearby, you will find a Korok that needs to reach its friend. (If you don't feel like helping the Korok right now, go northeast to Pit Cave.) To help the Korok, first look to the south to see the smoke signal. Then take a mine cart and attach a fan to the back, use Ultrahand to attach the Korok to the inside of the cart, and get in the cart and hit the fan. On the other side, remove the cart from the rails so it can't go back to the other side, then use Ultrahand and wiggle the control stick to detach the Korok and place it near its friend. Talk to the Korok's friend to get two Korok Seeds.
Go into the nearby stone structure and open the chest there to get some Zonaite.
Reach the Next Island
There is a slope near the Koroks with rails at the top, but if you look along the rails, you will see that they are partly broken. You will need to take one of the hooks from the ground and attach it to the middle of a cart, then hang the cart from the left (unbroken) rail. Make sure that there is a fan attached to the back of the cart, then jump onto the side of the cart and climb in. Hit the fan and the cart will go up to the other end of the rail.
Go down the slope here, check on the broken Constructs to get Zonai Charges, and go into the stone structure here. Open the chest to get a Portable Pot capsule. From here, you can jump down into the water below, where you can go northeast to enter Pit Cave.