Enigmata: Genu's Revenge is the sequel to the original Engimata, which also has medals. More of the same, along with some extra stuff.
As with all games of this type, there are a number of different strategies you can go by. queenofbabes wrote this about his strategy. Here's mine.
Missions are an invaluable part of getting achievement points. You get them and collect them by going through the portal that appears randomly in most levels, and they almost all involve killing lots of a type of enemy that occurs on the level in which you accept the quest. Some of the later missions involve enemies you won't see for a couple levels yet, and one involves collecting a particular rare drop, but those are the exception. Not only do you get cash and achievement points for completing missions, you also get some in-game achievements for finishing 5 and 15 of them, for even more. Finishing missions early will help unlock parts of Neko's Shop.
Focus on upgrading your base ship first. That will give you the best overall improvement early on. I like the blue ones, because I find them to be the most balanced.
Collect items, but try to save them as much as possible because you'll need them later. Genu and the final boss can all be beaten rather easily by stopping time and replenishing your energy while firing away, which means you need to save those purple items for as long as you can.
There are some rare weapon upgrades in the game, and they can be combined with specific normal weapon upgrades to get upgraded versions of the rare weapon. These combinations are:
unsheathed plasma (purple) goes with fusion
the blue orbs (forget the name) go with tetra fire
yellow and blue lasers go with double fire
The blue lasers with double fire are the most powerful weapon in the game. Each rare weapon has an even rarer variant which combines the same way as its base type does.
Nekon Attack missions are fun, but not necessary for the medals. Kill lots of stuff, collect all their drops, in a certain amount of time. Strategy for them I leave to the reader.
1) Cargo Expansion (basic). Increases your storage capacity so you can buy more upgrades.
2) Neko's Bonus Charm (basic). Increases item drop rate, which makes it easier to find things.
3) Gold Generator (elite). Gold is created as you fly, which makes it faster to accumulate enough gold to buy things.
4) INFINITE Bonus (black market). Your weapon upgrades last forever, or at least until you pick up one to replace it with. Invaluable for when you finally get that double firing blue laser.
5) Fiery Amulet (black market). More likely enemies will drop rare bonuses. Makes it easier to find the blue laser.
6) Time Nullifier (black market). This is specifically useful for getting the in-game achievements based around getting Amazing or Perfect in all 4 level categories; then you can replace it with something else.
7) Conquerer's Seal (black market). Boosts everything relevant all at once. Think of it as a minor overall ship upgrade.
8) Auto Neg. Effect Canceller (black market). Triggers your neg effect canceller as soon as you pick up a neg effect.
9) Second Chance (black market). When you die, you come back with rather a lot of shields and armor. Once.
Once you get to the point that you're using energy on a regular basis, you'll want to invest in the Megacorp V4 Nuclear AUX (elite) to get your energy charging as you play, and I like having the Micro Repair Bots (basic) also, which means all 3 of my ship stats are always increasing themselves as I play.
Beyond that, whatever upgrades fit your playing style. I always take the E-Magnet Missile Launcher (basic; missiles fire faster) and the Fion Assimilator (black market; unlimited missiles of all types) and all the extra attacker upgrades, because that fits with my playing style. That includes things like the Death Blast (black market) and Cytoreactor (elite).
The following skills are absolutely required in order to get through this game completely:
1) Healing Technique (basic). Trade energy for armor.
2) Destroy Shields (basic). Required for some of the early-mid levels where you encounter lots of a particular ship that has shields. After about level 15 you're highly unlikely to encounter those ships again, and can remove this.
3) Stop Time (rare). Time stops, you don't. You WILL need this in order to defeat Genu or the final boss, and I use it whenever I encounter some of the larger, more vicious ships on the way through the level.
4) Negative Effect Canceller (rare). When you hit a bonus with a negative effect, this skill removes it. Along with the Auto Neg. Effect Canceller shop item, this will keep you out of a lot of trouble as you play.
5) Bonus Lock (ultimate). Locks your current weapon so other bonuses won't change it. That means you can pick up bonuses indiscriminately and not worry about losing your double-firing blue lasers. You will have to enable this skill at the beginning of every level, and you can still lose your weapon if you pick up a bonus with a Z on it, as that is a bonus removing item that cannot be blocked.
queenofbabes has some other skill recommendations in his guide, which I will leave there as they are his. They come down to his playstyle, which may be different than yours. He plays a more skills-based game, while I play a more weapons-based game.
Never buy missiles. You don't need them, and only the bosskillers are even worth it. Instead, accumulate 11 million gold. Move Nukes and Death Missiles (both bosskillers) onto your skill bar and then purchase the Fion Assimilator (elite) to get them both to have unlimited ammo. I use Death Missiles on regular enemies, and then switch to Nukes for the bosses.
Here are the Rank 3 achievements decoded:
Warlord (200,000 Gold) - Heartlessly defeat 2000 enemies
Mega-Insanity (100,000 Gold) - Crash into a boss 15 times
The Box of Doom (140,000 Gold) - Collect an exploding box.
Filthy Rich (300,000 Gold) - Obtain 5 million gold
Strange Irony (430,000 Gold) - Destroy Neko's shop
Obliviator (400,000 Gold) - Defeat over 95% of enemies in a level past level 10.
Non-Random Achievement (800,000 Gold) - Just kidding, it was random.
Just Made It (1,200,000 Gold) - End level with 5% armor and shield remaining.
Titan (1,500,000 Gold) - Create a ship with 5000 armor
Sniper (1,000,000 Gold) - Finish a level with over 95% accuracy
Badge of Conquest (1,000,000 Gold) - Complete 15 missions
Full Capacity (2,000,000 Gold) - Acquire all skills
O.C.D. (2,500,000 Gold) - Own everything in the three shops.
Perfectionist Elite (3,000,000 Gold) - Finish a level with 4 Perfects beyond level 16.
Ultimate Defender of the Enigmata (1,000,000 Gold) - Defeat 10000 enemies
Box of Doom will drop randomly in a level above 15. Titan requires the Ledgendary [sic] Key, which I got as a random drop in level 19, in order to purchase the black market shop item The Fifth Collection, which gives you the Pseudo Epic ship, with a base armor over 1000. Then just add on every armor boosting upgrade you can find. The rest will occur as you play, with Strange Irony happening on random level replays.
Eterna Medal (10) - Defeat the ship who destroyed the supply base. Level 1
Mega-Goliath Medal (25) - Destroy the indestructible mega-craft. Level 10
Genu Returns Medal (100) - Take down the ulimate ruler of the universe. Level 20. By using Time Stop and Micro Energy Boosts (which appear to be bugged to restore all energy rather than only 50) you can take down Genu before he even gets a shot off.
The Entity Medal (100) - Beat the most powerful mega-craft ever created.. Level 25. I got this with Time Stop (used a lot but not constantly), lots of shield boost items, healing skill, and a lot of patience. This will be a 5+ minute fight.