Spent about an hour playing Athena today trying to figure it out and i'm finally starting to make it up to the first boss fairly consistently.
Unfortunately this seems to be about as far as i can work out the game so far, so i went off to GameFAQs to see if there were any guides...Sadly none...So i checked out Youtube instead and found this playthrough video from Youtube user ChangeV of stages 1 and 2.
It also includes a secret which allows Athena to increase in size and gain a flaming sword.
If you check out ChangeV's channel you can also find a playthrough of stages 1-3 for the NES version of the game.
I'd probably recommend trying to figure out the game before watching this video though, as knowing all of the secrets right away may spoil your enjoyment of the game a bit.