After your daring escape from the castle jail, you find yourself in a strange place.
Go south to go outside.
Go to the Dome
Go south and enter the dome.
Talk to the first person you see there and buy better equipment. Heal up in the Enertron, then exit the dome and go to the ruins to the north.
In Lab 16, don't touch the rats or they will steal from you. In battle, defeat Meat Eaters first, because they will repeatedly heal themselves and the other enemies.
First, go east, then north past the rat to find a chest (Berserker). Then go northwest and follow the path. There will be another chest (Lode Sword). Stay near the north wall to avoid getting attacked. Keep following the path, and go into the little crevice where the rat is running to find another chest (Lode Bow). Stay near the south wall of the next area to avoid fighting, and try to maneuver above the next monsters if you want to avoid them too.
In the next area, stay near the south wall and go south and a little to the west to find a chest (Ether), then go east. You can avoid any fights by going to the debris just south of the red monster and staying near the wall south of the red monster as you go east. If you fight Shadows, use Crono's Slash, Lucca's Flame Toss, or Fire Whirl. Go east, then stay near the west wall as you go north if you want to avoid battle. Keep going north to exit the ruins.
Go east and enter Arris Dome. Follow the path to reach the people, and there will be a brief cutscene. Go west and you can buy stuff from the kid behind the boxes. Use the Enertron, and save at the sparkle. Then approach the ladder southeast of the Enertron and there will be a brief cutscene. Go down the ladder. From there, go to the northwest corner of the room and go up the ladder there. Make your way to the northeast of the area, then go north and you will be attacked.
First defeat the bits, then attack the Guardian, because if the bits are alive when you attack the Guardian, they'll counterattack. Just use normal attacks. Keep your HP above 50. When the countdown reaches 0, stop attacking the Guardian. The bits will be revived. Kill them again, then resume attacking the Guardian.
Back to Arris Dome
Go north. After the cutscene, check on the guy to read his note, then open the chest (Mid Ether). Go back the way you came until you see a rat.
Chase the rat by holding down B, then pressing A when you're next to it. If you don't get it, just leave and re-enter to try again. You'll get a password.
You may wish to go back up to the Enertron to heal up and save. When you are ready, go back down the ladder, walk up to the sparkling console and enter the password. When you've gotten it to work, the passage to the door will open. Go through.
Go north past the rats and robot to find a chest (Mid Ether). Then go south and go up the stairs on the left. Follow the path past the bugs to reach a door in the northeast of the room. Go through it.
Go north up the stairs beyond the robot, then when the path splits, go northwest up the stairs and go through the door at the end. There will be a cutscene. Say okay when asked.
Afterward, go back the way you came to reach Doan and the other people in the dome. There will be a cutscene. Use the Enertron and save the game.
Go outside. There is a sewer entrance to the east, but you can skip that for now and come back later. If you do go through, you can meet Belthasar, who says some strange things. If you've already played through the game before, it's interesting to talk to Belthasar, but if this is your time playing the game, you might want to skip it.
Go north to Lab 32.
Open the chest (Mid Tonic), then go north. Check on the vehicle or try to go east. Then when you're ready, check on the vehicle again to start the race.
Use a B button boost when you're close to the finish line. Check how close you are to the finish line at the bottom of the screen. It also helps to try to get in front of Johnny so you bounce forward off of him.
Or, you can just lose the race and go east. You'll have to fight some monsters, but there is a chest in the area that contains the Race Log.
If you win, go west and fight some monsters, then open the chest to get the Race Log. Then, talk to Johnny and you can keep score of your race times if you race again. Finally, go south from the finish line area to exit Lab 32.
Go east, ignore the factory for now, and go to south to Proto Dome. Fight the monsters, use the Enertron, and go to the north part of the dome. Check on the robot and there will be a cutscene (default: Robo).
Now you have to choose who to leave behind. You might want to leave Lucca behind so you will have Marle's healing magic. You can always go back and switch party members later. When you are ready, exit Proto Dome and go north to the factory ruins.