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 to the southeast part of the forest, and look for the sparkle to the northeast of the monsters. Check on it to get it. For advice on how to use tabs, read the Guide to Tabs and Stat Growth.
To the west and south is Porre. There are no random encounters while walking around in the overworld, so feel free to go there if you wish. The Mayor of Porre 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, north of Crono's house. 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 (if you talk to it twice, it runs away, but you can just leave the area and come back to make it reappear), 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. Go all the way north. Marle will ask you to stop so she can get some candy. Don't move or push any buttons while you wait.
When she is done, go north up the stairs.
When Taban is done talking, talk to Lucca. You can either step into the Telepod on the left, or you can save time by talking to Marle to skip using the Telepod. When Marle is gone, pick up her pendant and there will be a cutscene and you find yourself in a strange place.