The first planet to face, Corneria is fairly easy and straightforward throughout the level. There are two bosses - by simply going through the level you automatically will end up facing the easier of the two. When this boss is defeated, Star Fox will proceed to Meteor; however, to advance to Sector Y, you must first get to the other boss by shooting down all the enemies targeting Falco at the point shortly past the checkpoint, then fly through all of the arches in the span of water past that. Earn the medal by keeping all of the team alive and in action and destroying 150 enemies.
To defeat the easy boss, simply shoot his legs until he cannot move, then shoot his back until the shield meter empties entirely.
In order to destroy the more difficult boss, follow Falco through the waterfall and shoot down the enemies you see. Duck down when you reach the water and then begin to shoot at the boss ship's arms as soon as you can. Wait for the arm to open, then shoot directly into where the enemies are coming out (don't worry about them). Do this with all three arms until he turns away and then continue to shoot and bomb him as much as you can; when he turns back towards you, watch out for the orange torpedoes.