-Taking damage on slopes sometimes won't propel you into the air, but I don't know why. Usually moving around to a lower part of the slope solves the issue.
-The first room: getting the Life Capsule is essential. Try to get sixteen points here, I can never get fifteen.
-It's possible to finish every part after Monster X with zero jumps. It might sound difficult, but it's certainly doable.
-For Monster X, you can take damage on the red barrier while riding it to go to the other side without jumping and turn it around. You only take one damage from doing this.
-During the Monster X fight, try keeping a Gaudi alive (I think keeping the flying one is easier, and its projectile only does three damage) so you can skip the jump after the fight.
-I skipped only one Save Point, and that was the one in the Labyrinth Shop. Overall, you'll save about four jumps and it's the easiest to skip, so I recommend it.
-Keep the Blade at Level 2. Well, this tip in specific I follow for almost every Cave Story run I do, but whatever.
-I advise equipping the Bubbler when tanking damage, it's the least useful out of all the weapons you have.