Get good at dashing, jumping, and wall jumping precisely. You'll have to do all three at once and understand the trajectory of your jump if you want to avoid attacks and be in a good position to dodge the next ones.
Sure, you can win by exploiting elemental weaknesses, but you can win with the X buster too. I personally thought Storm Eagle was the easiest to beat without weapons. Also, shoot homing missiles at bosses if you have time but aren't in a position to hit with anything else. I like to switch weapons by pausing, no sense in wasting game time cycling through them.
If that doesn't help, you can always get the hidden upgrades scattered throughout the levels. I didn't know about these and didn't explore much, so I actually beat the game with stock equipment. Take care to recognize the boss' tells and you'll be alright.