While there's no set order in which to do most of these stages, this is the order of stages in accordance to their bounties.
Skies of Syd - This is the first stage. The Drone ships only fire straight ahead, so dodging them as well as their bullets shouldn't pose much of a challenge.
Asteroid Field - You'll start to see some new enemies, namely rocks and Swerve enemies. It is highly recommended you do not run into enemy rocks. They will damage you badly.
Fortress City - Somewhere in the middle, you'll be presented with a wide open missile field. Surface to Air Missiles accelerate quickly, so they are harder to dodge the longer they travel.
Great Forest -
Coral Ocean -
Acid Swamp -
Frozen Tundra -
Hidden Ruins -
Earthwall - Refer to the Boss Strategy section on how to battle the Earthwall.
Federation Skybase - This is a very simple stage, with nothing but Drones. So if you find yourself stuck, you can alawys come back here to grind.
Starship Graveyard - World 2 as a whole is largely straightforward. You'll be picking up various multi-hit weapons. As long as you keep on top of those, you should fare pretty well.
Neon City -
Jungle Maze -
Forest of Life -
Ocean City - The water pressure has increased, so consider investing in more engine mods.
Swamp of Lost Souls -
Icicle Cavern -
Forgotten Ruins -
Firebolt - Refer to the Boss Strategy section on how to battle the Firebolt.
Gravstone Fields - World 3, things start to get hairy. The enemy density is much higher, and bosses have new tricks up their sleeves. Strongly consider buying and upgrading your Area of Effect weaponry, as they will help you get through many stages.
Remains of Lucerna - You'll notice there are many, many rocks in places you can't ordinarily shoot. Strongly consider coming back with a high level Wave, Charge, or Chain to deal with those rocks. Don't be afraid to blow an Overdrive to plow your way through something. Remember, you can fly through terrain during Overdrive!
Forbidden City - The large volume of rapid fire sentries makes this dangerous to traverse without the aid of piercing weaponry such as Charge, Wave, or Chain. Once you've figured out a way to handle all those pesky buggers, the rest of the stage isn't too difficult.
Death Jungle - This is a fairly straightforward stage, but beware missiles rising from the bottom in tight areas. Missiles generally hit pretty hard.
Spiral Forest - Similar to Death Jungle. It's fairly straightforward, but watch out for crammed areas.
Maelstrom - The water pressure is considerably high here. You may wish to max out your engine mods before entering.
Eternal Swamp - First thing you'll notice is... no terrain! However, this is a stage with nothing but enemies. Consider using your best Area weaponry, such as Scatter. Crush can be surprisingly good here, as there is no terrain you need to pierce.
Absolute Zero - Pay attention to the terrain! If you do not move fast enough, you will be cornered into a wall and crushed to death. Don't be afraid to use Overdrive if you're in a jam. Other than that, this stage doesn't have too many enemies, and isn't too problematic.
Ancient Ruins - Very similar to Forbidden City, but with some tricky spots thrown in, particularly the missile field in the middle. Hopefully you have a good Chain by now. Barring that, Charge, Wave, or even Sawmines might see you through. But for the boss, you'll likely want Charge...
???? - Refer to the Boss Strategy section on how to battle this.