Typically the top right room is done first, because it requires a lot of eggs, and keys take up egg spots so you want to do it before getting any keys.
The bottom right room has an extended flutter trick to skip riding the slow arrow. The difficulty with this method is clearing out the spikey headed enemy, who cannot be used for an extended flutter. After that, the key can be obtained quickly by using a perfect flutter to immediately grab the key and go through the door.
|4-4 Key Grab|
The top left room is fairly straightforward, though it can be difficult to clear the foam quickly.
If you are out of eggs in the bottom left room, eating one of the small flames can be used to break the key cloud. Check out the animated gif below for an optimal Marching Milde fight.
|4-4 Boss Fight|