Since your next destination is in the desert, first return to Hyrule and go to Ravio's shop. Rent (50 Rupees) or buy (800 Rupees) the Sand Rod since it is required for you to continue in the desert region. After that, head south from your house into the Great Swamp, then follow the path leading southwest into the desert. You should find a fissure portal there; merge in and pass back into Lorule at that point.
Misery Mire/Desert of Mystery
Upon your arrival in this damp area, you will find that Misery Mire is again separate from the surrounding area. Follow the path to the southwest and you will enter the swamplands. Go up the nearby steps and activate the weather vane right away, then head north to another fissure portal. Return to the Desert of Mystery in Hyrule, then go southeast and up the steps onto the plateau with a small pool of water. Use the Sand Rod to form a bridge westward to a pillar with 20 Rupees atop it. Continue west via the Sand Rod and merge onto the wall to reenter Lorule via the fissure portal.
Next, follow the bridges leading to the south and west while defeating the enemies along the way. At the end, use the next fissure to return to the desert once more. You need to cross a series of pillars by using the Sand Rod here. Go east and across to the second column, then go north, west (grab 20 Rupees), and north to the wall. Merge with the wall and go west (take 10 Rupees along the way) onto the solid platform ahead. Go up the steps and activate the "Desert Palace" weather vane, and save while you're here. There is a hidden Maiamai just to the south of this platform if you use the Sand Rod on the ground to dig it up.
When you are ready to proceed, stand north of the weather vane and use the Sand Rod to form one last bridge. Use it to enter the Desert Palace.
NEXT: DUNGEON: Desert Palace
