Defeat Bat Guy
Stand on top of one of the two platforms, and then hit him with jumping slashes when he is clinging from the ceiling. The bat guy himself isn't much of a problem provided you stay out of his pattern. If he clings on the ceiling and remains still, go for it. To get rid of the bats, try to hit as many as possible with jumping slashes, being careful not to land on some bat coming from below. You can also try shooting at them from a distance. If the screen gets full of bats, use the magic to wipe them all out.
Defeat Frog Riding Boss
Don't let him trap you in a corner, or you will get burned. When he's on the frog, wait until he lands from a jump then get in close and hit him with two quick jumping slashes (both in the same jump) to knock him off. You can also get moderately close so the frog starts spitting fire, then retreat and hit the rider with projectiles, but it's much faster using the sword, and you'll also build some magic. When he starts bouncing all over the screen it's better to always try jumping slashes as they are safer and deal twice as damage than standing slashes. You can also try using magic. Also, on the very top of the stage there is a health item (in Easy and Normal modes), you'll have to use your rope to get it (preferably when the boss is on the frog so he won't get in the way).
Defeat Bat Guy...