The following is an in-depth guide for the second area of Metroid II: Return of Samus.
Keep heading right until you inevitabley find another upward shaft. Go up it. Then go right through the passage that follows. Here you'll find another shaft. At the top you can go right into it to reach another hallway. Just keep heading right until you eventually find a downwards tower. There should be a save point you can use right before it. Go down this shaft. At the bottom, go left and you'll find an upgrade, the bombs. This will help you bomb through any path in your way, which is something you'll likely have to do. If there's any time where you need to get through a wall, you'll likely need to bomb somewhere around there. Head back right to the shaft but this time go into the first right exit you'll find and immediately destroy the shooters and roll and bomb above their heads to enter the wall. You might need to bomb a few times to get to the other side. At the end of this you'll find an energy tank which will give you more health as well as fully heal you on the spot. Reach the next shaft to the right and go down and through the first right door. Here there should be three missile expansions. One you can't get, but if you shoot the breakable block from above and jump at the right time so the block respawns and pushes you upwards you can get it early. Otherwise come back when you have the Hi-Jump Boots or the Spider Ball, but you'll have to remember yourself. Go back left through the wall and then down to the first door on the left. Keep going left and you'll find the Ice Beam, which will do less damage but will allow you to freeze enemies, making traveling the world easier. Go back right to the shaft, but this time go up and through the second door on the right this time. Keep going to the right until you can't anymore, and go down the brown yellow spot in the ground; it isn't solid. Use you're bomb and morph ball to boost you around but eventually you'll see the upgrade through the walls, just bomb it and go through to get the Spider Ball. If you press down a three times when standing, you'll enter the Spider Ball which can cling to ceilings and place bombs. You can use this to escape this hole. Go left until you reach an area with a thin bridge and no ceiling. Use the new Spider Ball to climb the right wall and get up to reach another right door, above the previous one. Go through here and kill the Alpha Metroid inside. Go left from the area with the thin bridge and go up the shaft and to the first door on the left. In here you can get two missile expansions. Keep heading left bombing through the wall to get past. Keep heading left through the doors, you might have to fall down a bit, and you'll reach another shaft. Go all the way to the bottom and go right. Here, head right all the way until you are forced to fall down; fall down. You can get a heal pool in that first room if you need it but it's not necessary. Here, head right and keep heading right until you find an Alpha Metroid, kill it and go back. Go left and down the shaft, then go left and kill the Alpha Metroid, then go all the way right and kill another alpha metroid. Then go left until you find an upward shaft, go up, then go right a tiny bit until you find an area to go upwards, then go all the way left until you find a shaft. Go up and through the first door on the left. Keep going left and then down the next shaft and left. In this room keep going left until you find a hole in the floor, go in to enter Phase 3.