This one can be a bit challenging due to the dodgy AI and controls.
Firstly get used to the controls so you can move around properly. Once you feel comfortable start the game proper and kill all the enemies in the first area and collect all the stars. Take any door it doesn't matter. At the upgrade screen choose the shields.
In the second area, again shoot everything and collect all the stars. The key is half way through the level. Take the door and pick shields again and longer range bullets if you have enough.
This is where it gets tough in the third area. Try to kill all the enemies without getting hit much (longer range bullets help here). Once you reach the end there should be a base. Enter the base and then leave before it blows up. The trophy should pop then.
This game isn't particularly confusing but the controls are terrible and take getting used to. Picking shields as your power up will mean you take lots more hits so I'd recommend doing that.