Guess because I play lots of retro games I'm more used to the higher difficulty of the arcade games, usually I don't aim to lose any lives on the first couple of levels of Fantasy Zone or Space Harrier, that way I have a stock of lives for later levels where it gets harder.
Never stay in one place, enemies will shoot at where you were, so as long as your moving they won't hit you. Stay off the ground, keep in mind you can shoot trees, clouds etc. so I usually fly around the top half of the screen. Memorise the enemy attack patterns in the early levels so you know what each are capable of, and play around with the controls - analog stick and D Pad each have your character control slightly differently so use whichever suits you best. Also keep in mind that the triangle button inverts the Y axis controls.
I think any of the Space Harrier challenge scores can be achieved by clearing world 5 (the bonus stage), killing bosses gets you more health as opposed to letting them disappear, the second boss is a pain as the enemies shielding the boss will fly away as soon as you kill the central enemy so try to leave him until last if you can.