After starting a new game, you'll see a stunning cutscene showing Zack Fair (the main and controllable character) jump onto a moving train, and take out a bunch of soldiers in stunning fashion. Once the cutscenes have ended, you're faced with a bunch of soldiers that you have to take out yourself.
Fortunately, you get a brief explanation on the controls of the game before you start. Once you've finished reading, take out the Sentries by either Attacking, or using the elemental spells Firaga, Blizzaga and Thundaga. If you prefer physical attacks, then use Assault Twister, a wide reaching physical attack that will hit lots of foes at once. This is also an opportune time to get used to dodging and guarding enemy attacks, as even if you do get hit, you only take 4 damage at the most.
Of note is the spinning slot machine in the top left hand corner, called the DMW (Digital Mind Wave). This device allows you to level up yourself and the materia you have, as well as summon and perform powerful physical attacks. However, this all depends on whether it matches up three of the same slots.
After defeating all of the enemies, you will be prompted to go up the stairs, which you will do in a cutscene, to face the first boss of the game.
This guy can be tough if you don't know what you're doing. As with other Behemoth's throughout the series, it knows the heavy hitting attacks Heave and Claw Swipe, as well as a new attack Gyro Tail. If you're not careful, these attacks can kill you very easily. And if you are trying to attack from the front, you'll most likely be guarding more than attacking. Even if you do get a chance to attack, you won't deal much damage.
The best course of action is to attack from behind. That way, you will be able to do automatic critical hits, and the Behemoth won't be able to attack you that easily. If you do manage to get hit, you can always use a potion. Also, make use of the materia that you have - it'll be gone soon.
After the boss is defeated, a cutscene will start, and the chapter will end.