I managed to do it with just Marines, Medics, Marauders and Siege Tanks. I put up two bunkers and four tanks at each ramp for defense, then built up my forces for the assault. I attacked the base directly to the north. The initial tactic is to place tanks in siege mode a little bit back from the protoss ramp so they are out of range of photon cannons. Then run your MMM ball halfway up the ramp so the tanks can start shooting down the cannons. Meanwhile keep the drill hotkeyed so that it can assist in taking down archons, collossi, void rays, and anything that's doing damage. Don't let your MMM ball run out of reach of your tanks.
When you've beaten back their initial defensive line you've got a foothold. Move your tanks up and keep on pressing in. Focus on taking down the pylons around unit producing buildings and keep your drill handy to zap the occasional pylon or counter-attacking heavy unit.
The real key to this strategy is you need to keep reinforcing. Keep your barracks and factory hotkeyed and move the rally point to deliver troops near the front as you move from left to right across the map. Before you've taken out the entire northern base you should have the achievement. Good luck!
Disclaimer: This is very much based off of this Steam forums post. I didn't write it, but I've tried to expand it a little, based on my own playthrough to get the achievement.
First of all, you do need to beat The Rise of Africa before you can do this. It shouldn't take you all that long, and you'll get a (very!) basic feel for how the game works.
After that, play the Fuel Crisis scenario. Start by recruiting a single agent in all regions except Africa. Create a Regional Technology Office in each location, and end the turn.
On the second turn, replace agents that have gone missing, add an agent in both Africa regions, and give them a tech office too. Set Japan to research Energy, and create a Political Office everywhere else. End the turn.
Keep Japan on Energy research until they discover CCS. Everywhere else, try to keep your hearts up so they don't ban you too early, and when you don't have anything else to do in a region, have them commit to nuclear energy. You will also want to play the Black Ops card in the more stable parts of the world; this will help you later.
When Japan gets CCS, have everyone else acquire it. Keep Japan on research.
The next turn, you should get 4th generation nuclear energy in a couple of regions. For those regions who have not gotten there on their own, acquire it.
From here on, simply commit to nuclear everywhere. In the more stable regions, play Fund Insurgency Campaign to reduce stability, then commit to nuclear. Keep doing this, and hopefully, you will quickly reach a global thermonuclear war.
If you fail, try again - there's a fair bit of RNG involved in this game, so you might get banned too quickly, or miss the target before you cause enough instability.
TL;DR: Get tech offices for everyone, have Japan research energy, import CCS and 4th gen nuclear everywhere, commit to nuclear and reduce stability.
Best Answer
Fireworks are triggered when a knight returns a princess to a castle. This means you kill a carriage and don't take the princess.
I used the first HOARD level, because it has the longest expected goal times (so they must expect you to survive longer). I started out each round by killing as many towns as possible, and keeping the archer count down afterwards. Towns give you a nice initial gold boost.
Whenever a magic tower springs up early on, killing it will give you lots of gold (and likely upgrade your dragon). They're also the most annoying thing on the map late-game (second being archers).
Make sure to use powerups - they're great for picking off groups of knights and individual magic towers. As for archers, you can run by them and not die as long as you don't stop.
Once you're good at simply surviving, you have two secondary goals:
Kill princess carriages, and let knights take the princesses. When a knight returns a princess to a castle, that triggers fireworks. You need to do this twenty times.
Don't get hurt. If you get hurt, you need to take a princess yourself, and that's one less firework :)
It took about a six minute round to get the achievement, after about 45 minutes of practice (and the game was fun enough I didn't mind the practice). When it gets that far into the game, you'll start seeing two or three carriages together you can kill for fireworks.