Chrono Trigger/Beyond the Ruins

Chapter 6: Beyond the Ruins (2300 A.D. first time)

Trann Dome and Lab 16
You'll start out in the Bangor Dome, which doesn't have much besides another sealed door. Buy some new armor and use the enertron, then head into Lab 16 to cross over into another area. Try to avoid the rats, they steal tonics and money. Head to the right to pick up a beserker accessory, and make sure you grab the lode bow for Marle. There will be some ghosts in the next area who you'll need to kill with Lucca's fire attack. Try to get clumps of enemies with fire whirl as well. Once you've fought your way past some mutants and more, you'll reach another outdoor area, where you can access the sewers, Lab 32, and Arris Dome. The sewers aren't strictly neccessary until much later in the game, and you can't go through Lab 32 until you complete a quest in the Arris Dome, so head there.

Arris Dome
Talk to the people inside to learn a bit more about the state of things in the future. When you're ready, head down the ramp to explore a new area and get into a boss fight. Walk through the girders and exit to the north to hear an alarm go off in the room you walk through. You'll have to fight a Guardian boss which has two bits. The bits periodically attack you, and if you attack the Guardian itself while the bits are active, they'll do a special attack. Concentrate on killing the individual bits, then attack the Guardian while it's counting down to regenerate the bits. Use Aura Whirl with Crono and Marle to restore health during the countdown if you need it. After a few rounds it should fall eventually.

You'll get a seed and learn a clue about the rat running around. Try to catch the rat if you want, but the clue it gives you is to hold L&R and press A. Use that to activate a bridge to the right side, and fight or sneak your way through to eventually reach a computer. Activate another bridge to a sealed door if you want as well. You'll learn about the nearest Time Gate to the east of Arris Dome, and watch a recording of the day of Lavos, which explains why the future is so apocalyptic. You'll make it your official quest to change the past and try to stop Lavos from destroying the world. Head back to the top of the dome and you'll give the seeds you found to the people living there. You'll get a bike key, and you can use it to head into Lab 32, and race against Johnny on a jetbike.

To race Johnny, you just need to be ahead of him at the end of the race, which should be easy with your three boosts. If you choose to walk the highway instead you can find a race log which will enable you to record your bests speeds and compete for prizes. You increase your speed during the race by boosting or having johnny's bike bump you from behind. If you pass a speed of 1500 you get a power tab.

Once you pass through Lab 32 one way or another, head into the dome, fight a few buggers, and meet Robo. He'll join your party and tell you he needs to go to the factory to turn on the power so you can reach a new time gate.

Challenging Lavos
Impossible during this stage of the game since you have no access to a time gate.

Continue to Chapter 7: The Factory Ruins (2300 A.D. first time)