Go through the Mimiga village and collect the Map item, then when you have defeated Curly use it to find a secret passage in the back room in the Sand Zone. Use the Map to reach the end of the passage and use the 'Look' command on the ledge on the left to get Curly's underwear.
Keep your Polar Star and don't trade if for the Machinegun. You will need the Polar Star and another weapon later in the game to trade for the Spur which is the most powerful weapon in the game. This weapon is also refered to as the charge shot as there is no need to collect weapon energy to power it up as all you need to do is hold down the firing button to charge your blast.