The first eight levels that you'll have to go through serve as a pretty good tutorial for what's to come. Below are some tips and other bits of information to help you through, as well as some info on how to obtain some of the earlier achievements of the set for this game.
For the first stage, just try to get comfortable with how the game works. Enemies will pop in from the sides of the screen in waves, and when all of them are in position, one enemy at a time will soar towards you and open fire. Try to watch out not only for their attacks, but the ships themselves; getting hit by enemies isn't exactly a great tactical strategy for defeating them, and you'll lose a life in the process.
This is also the best time to get one of a few miscellaneous achievements. Take note of those green enemies that sit at the top of the screen and turn purple when hit once. These are Boss Galagas, and they're related to quite a few achievements in the list. For a later run of Stage 1, try to shoot all four Boss Galagas until they turn purple, and then shoot enemies as usual. Leave about six or so around the stage and let the Boss Galagas swoop down and use its tractor beam. You're going to want to be directly under it as it comes down onto you, and you're going to want to shoot right before you get sucked into it. This will technically put you inside the tractor beam as the Boss dies, and you'll unlock the below achievement for escaping its tractor beam.
If you decide not to shoot before you get sucked in, you're going to have to try to hit it as you're spinning around in the beam, which is ridiculously hard to do and is incredibly luck-based. It's best to just go with the trigger finger method mentioned above.
Enemies will now fire at you upon entry into the stage. Be careful about moving around. It's actually a better idea to focus on not getting hit instead of aiming at enemies sometimes. Additionally, two enemies will drop down towards you once all of the fleet has been assembled, as opposed to the previous one from before. Try not to put yourself in a corner; they might seem like safe spots, but backing into one could mean death since you won't have much time to move out of the way of an enemy shot.
This is your first Challenging Stage. None of the enemies will fire at you in a Challenging Stage, so this is purely a break from the action to rack up points. In later Challenging Stages, you'll have to defeat all of the enemies in each wave, but we'll worry about that when it's time to.
Things get even crazier from here. Up to three enemies will come hurtling towards you, and will fire more shots as they do. Although death has a much weaker sting with the stage selection option in the menus, it's still good to build up practice avoiding enemy fire. After all, this is just the fourth of 32 Stages.
You should also spot a special scorpion enemy during this stage if you refrain from shooting the yellow bee enemies for too long. Merely seeing it will give you an achievement, so don't worry about trying to shoot it. If you didn't find one on this stage for whatever reason, you'll surely find one later. And if you still haven't found one before Stage 8, then don't worry about it; you'll have a 100% chance of finding one on Challenging Stage 15. If you want to find one now, though, check the description for Stage 8 for more instruction.
For this Stage, we'll be aiming for a certain achievement. Just make sure you have at least two lives for this, because you'll have to lose one in the process.
During the level, try to get your ship caught in a Boss Galaga's tractor beam. Don't shoot while this happens, because you'll want the Boss Galaga alive. When it has one of your ships (now red) at the top of the screen, shoot the Boss Galaga to death. Make sure the Boss Galaga isn't flying around, because that'll create the scenario for the achievement you'll be aiming for in Stage 6. Killing it while it's idle should cause the red ship to attack you like any other enemy. Defeat it to earn a fairly simple achievement.
You'll get a bit of a break for Stage 6. Enemies won't fire at you when they spawn in, but that doesn't mean you shouldn't fire back. This is another good chance to get a new achievement. Hop back into a Boss Galaga's tractor beam (remember to have at least two lives) and don't shoot it down like you did before. Instead, destroy the Boss when it's flying towards you. This will prevent the red ship from turning against you and you'll now have two ships flying side-by-side to fight the enemies. Using two ships is going to be a big part of the strategy for Challenging Stages later on.
Another Challenging Stage. You won't be able to die on this one and you don't have to worry about killing absolutely everything, so it'll be smooth sailing to get to Stage 8.
This is just your average level, but a new enemy will spawn out of the bees if there are few enough of them left on the map and there are little to none of the other types of enemies on screen. If you want to surgically remove all the other enemies and leave about ten bees behind, one of them should turn into a set of three stingrays, just like they would with scorpions in Stages 4-6. You don't need to find a stingray right away, since these too appear in a later Challenging Stage.
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