Fort Dragonia
You'll begin Chrono Cross with a party of three - Serge, Kid, and another, randomized person you can get later in the game - raiding the heavily-fortified Fort Dragonia. This short introductory area will make you familiar with the basics of combat, but provide little context as to what's going on.
Fort Dragonia is a pretty quick jaunt. Take a right from the elevator and go south. You'll emerge in a large room with an inactive elevator in the middle, out of reach. Follow the path south and under a bridge to find a doorway. Take a left in the next room and go up the stairs. The next door will take you to a higher platform in the large room, near the elevator. Wander past it on the right and go north to find a room with a crystal. Touch the crystal to activate the elevator, then backtrack to the large room, take the first set of stairs on your right, and get on top of the elevator. Confusing cut scenes will follow, and your brief foray into the... future...?... will end.
(There are enemies to fight in Fort Dragonia, but you're more than powerful enough to mop the floor with them. You're better off just skipping them, as this section has no bearing on the wellbeing of your party.)
Arni Village
Serge will wake up, alone, in Arni Village, a small fishing community. Home sweet home. Check under Serge's bed for 200 g, and look in a pot on the left side of his room for a Tablet Element. You can equip this on Serge by Allocating it to him via the menu, if you so desire.
Speak to Serge's mom for a bit, then leave the house. You need to find Serge's childhood friend, Leena, but you might as well look around a bit first. To the left of Serge's house is a man, a kid, and a big fish. Speak to the man and you'll receive a Komodo Scale by indulging him and saying 'You're probably right'. If you continue to the west you'll find Radius, the mayor of Arni Village, and he'll teach you the ins and outs of combat in Chrono Cross. If you check the hut on the left side of this area you'll find an AquaBeam Element in a bucket out front and a Cure Element inside, behind a wall scroll on the left side of the hut.
Check the center of the village. There's a woman selling a scant number of Elements here. She gets outclassed in terms of her wares very quickly by another vendor, but what she offers isn't bad for the beginning of the game. Check her cart for the Shellfish Custom Frame. Just north of the vendor is a boy idling beside some huts; use the Komodo Scale as a key item and you can either give it to him for free or sell it for 1,000 g. He'll give you an Uplift Element if you hand it over for free.
There are two huts to check near this boy. Look in the upper hut and you'll find a man in his basement, wondering what could have been if he'd done things differently in his life. He'll give you a Shark Tooth. You need this for recruiting someone later, so don't forget it. The lower hut in the east is a restaurant, and you can look in the back room (far right side of the screen) to find a Heckran Bone under a sleeping pallet and an Ivory Helmet in a chest. Speak to the waitress in the restaurant about poems - useful for later in the game - and check the barrel near her multiple times for the Uplift Element. (What's with the barrel, exactly?)
Poshul
If you want to make the first 'dungeon' of the game a fair bit easier, head to the field where Radius is lecturing, in the west of Arni Village. You'll see a big, pink dog gallivanting about in the background. This is Poshul, and if you speak to her with the Heckran Bone in your inventory she will offer to join you on your quest. If you don't recruit Poshul now you'll have another opportunity in the near future.
With all that done, head to the pier in the north of the village. Leena is here, and she chastises Serge for being a lazybones. He has a promise to fulfill: Collect three Komodo Scales. You'll need to head to nearby Lizard Rock.