After starting a new game, Crono wakes up in his house.
Go to the Fair
Go downstairs and talk to Mom. You will be prompted to enter your friend's name (default: Lucca). Talk to your Mom again to get some money. Then leave the house.
Go to the mayor's manor to find a couple of chests (first floor: Tonic; second floor: 100G). Go upstairs in the mayor's manor and talk to the mayor twice to get more money.
If you wish, you can now go to Guardia forest to the west to gain money and experience. There is also a Power Tab in Guardia Forest. Go up and to the far right, just above the monsters, to find it. For advice on how to use tabs, read the Guide to Tabs and Stat Growth.
To the west and south is Porre, where the Mayor will give you 10g if you act like a chicken, and there is a chest upstairs in the Mayor's Manor (Shelter) as well as a couple of mysterious boxes.
When you're ready, go to Leene Square. When you go up the stairs, bump into the girl who is wandering around. When you stand up, first talk to the girl, then get the pendant and give it to her (if you don't talk to her first, there will be some consequences later). Let her join you, and name her (default: Marle).
There are a few good things to do at the fair. First, there is a cat in the area to the left of Leene's Bell. Talk to it once, then have it follow you to the area right of Leene's Bell, and talk to the girl there to return her cat. This will have good consequences later. Also, don't take the pink bag near the old man, because that is his lunch. Taking it might cause problems for you later. And one more thing: don't try to sell Marle's pendant.
When you are ready, talk to someone by the fountain and you will learn that Lucca's machine is ready. Walk all the way north. Marle will ask you to stop so she can get some candy. Don't move or touch any buttons while you wait.
When she is done, walk north up the stairs.
If you are playing a New Game +, there is a sparkle in the right-hand pod. Touch it to go directly to the final boss. If you win, you get the ending "The Dream Project." For a real challenge, don't touch the sparkle until Marle is waiting for you to try out the telepod. This way, Crono is the only person in the party, and you can try to face the final boss alone!
When Taban is done talking, talk to Lucca. Save time and talk to Marle to skip using the Telepod. When Marle is gone, pick up her pendant.