You'll begin the game on the surface of SR388 by Samus' ship. Remember that you can use her ship to refill missiles and energy. Go right to find the main entrance leading down into the bowels of the planet. He inside then proceed through the next tunnel filled with a number of creatures. You'll locate a large hole at the end of the corridor, jump down to find a shaft leading down.
Blast your way through the crumbling blocks connecting the walls of the shaft, crawl through the small tunnel at the bottom using the Morphing Ball. Shoot your way through more crumbling blocks and climb the next shaft, then head right.
Shoot through yet more crumbling blocks down the next shaft and drop down. Shoot down as you fall down the next massive shaft to blast through all the crumbling blocks. Once at the bottom you'll reach a large cavern with the first Save station on the right, so you might want to save the game now.
From here we'll go left to face the first Alpha Metroid. Go left through the next tunnel, then drop down the long shaft filled with flying enemies beyond the tunnel. Proceed through the next tunnel filled with enemies, once you reach the end you'll find the cavern containing the first Metroid. It'll be sitting on the ground, once you come close enough the Alpha Metroid will rise up out of its shell. Switch to missiles and blast away at it as it flies around the room.
Try not to take too much damage, once you finish it off you can find energy and missile refill capsules hidden on the left of the room. After a few seconds the screen will begin to shake and you'll hear a rumbling sound. This indicates that an earthquake's occurred somewhere nearby, lowering deadly lava from a cavern, allowing you passage. The hunt is on.