The Frozen Flame
Chronopolis is pretty much a big area to just explore. Make sure to inspect all floors, and don't move on without the Forget-Me-Not-Pot or StrongArm. The enemies are pretty easy to avoid, and are pretty weak anyways.
If you search a little bit, you'll find a little event where you must control a little cleaning droid. Use the robot to activate the bridge to cross to the other side of the water. Now, you can just shut the droid off and continue.
Now, you must find a way to deactivate Level 2 security. A nearby lab has digital maps of El Nido on the floor- This is the room where you can deactivate security. Head to the front of the room, and inspect the control panels. Now, you can head to Level Four and watch the meeting explaining Gates, a Chrono Trigger paradox. Now, you'll need to find a technician chief down at the docks. Time to head back down the ground floor. Once you've found the chief, you can access a room explaining the Records of Fate, and the Arbitor Keycard you need to get to the Frozen Flame. You'll find a computer journal of memoirs, which are rather interesting. When you're ready for Lynx, take the elevator to the basement with your new Key Card.
My Dear Doctor Ashtear
Just outside of the elevators, you'll need to beat up some sentry robots to get by. Blue elements work nice, and physical attacks help too.
At the end of the hall, Serge is recognized as the "Chrono Trigger" by the Chronopolis computer. Step through the door, and prepare for Lynx, the Frozen Flame, and the explanation of what on Earth is going on.
Lynx explains to you why he needed Serge's body to reach the Flame, and what FATE's mission is. He tells you that FATE was designed by Belthasar, and the scientific genius, Lucca Ashtear. (Who saw that coming? ^_-)
Explaining himself to be the living embodiment of FATE, Lynx attacks. If you're lucky, you may suceed in whipping off a Saints and put him to a quick death. FATE slowly counts down to his ultimate weapon, using weak black attacks each turn. Try to take him out before he pulls off his attack, but don't hurry too much. Serge, Zoah, and Karsh may be good choices; Zoah and Karsh can cause serious damage with their Duel Tech, and physical attacks. Zoah, with his awesome HP, should easily be able to survive FATE's ultimate attack, if he manages to pull it off. Good luck!
Masamune's Legacy
After you defeat FATE, Harle and Kid exchange words. Don't worry, I hardly understand what they're saying, too. When it's all over, you're treated with an FMV of Sky Dragon Isle rising from the ocean and forming what is later called Terra Tower.
Now, you can head to Hermit's Hideaway with Mastermune and save Kid from her past. Within the blade, Masa and Mune recognize Kid as "the princess", and offer to send your team back in time to save her from her sad past.
As you awaken in the past, you find yourself in Lucca's House, which is ablaze. If you want Kid's HotShot limit break, type L2, Triangle, R2, and Triangle into the machine near the next door. Head up the left stairs through the door, and get the Ice Gun from the back wall. Head back downstairs, and up the right steps. Check the square in the back of the room, and head into the secret passageway. At this point, Serge is the only person left. Put out the flames with the Ice Gun, and watch the following scenes. Remember to bring a tissue.
Starship Enterprise
After you've saved Kid, or if you decided not to, take Steena and the two Tears to Divine Dragon Falls. The two tears will form together, giving you the Chrono Cross item and element. Equip the element on Serge, and don't worry about it for now. Now, take Starky to Another World's El Nido Triangle. He rigs your boat with an antigravity device, giving you access to Terra Tower. Terra Tower is the final dungeon, so prepare yourself. This is a good time to get everything you want, and to explore some more.