Either weapon works; feel free to experiment. When the battle starts, check out the shacks and get all of the items that you can. When he gets close, run between his legs, get behind him, then open fire. When he is approaching you, use the shotgun and keep shooting him in the head while avoiding his attacks. When he roars and starts moving around, something will pop out of his back. Before it kneels on the ground, get in close until the action command that reads "Climb" appears. Use it to get on his back. The screen will indicate which button to use to slash. Press that button as fast as possible until you get off. Repeat the process one to three more times to defeat him. Additionally, when facing the El Gigante, buy a rocket launcher; it will kill him in one hit instead of wasting your other ammunition.
You will have to fight four El Gigantes (the big monster that breaks through the door and picks up and crushes Leon). You will fight one by yourself, one with the girl, and two at the same time by yourself. The quickest way to kill them is to buy a rocket launcher immediately before the fight so that you do not take waste space in you inventory. Throw a flash grenade to stun him, then run around behind and shoot him in the back. You will kill him in one hit. The same can also be done on most of the Bosses. If this does not kill them, it will do a lot of damage.
To win easily against the two El Gigantes in Chapter 4-2, go up the ladder and go down the rope on the other side of the room to activate the mechanism. It will open the circle in the middle of the room and take an El Gigante into the lava. Do not try it a second time; the mechanism will not work again. Just use the rocket launcher to finish the remaining El Gigante.