The guide
Solve it.
Then solve it fast.
Take your own cube and follow along. The cube beside this text shows every step: tap any move or algorithm and it plays, slowly, one turn at a time. Pieces that don't matter yet turn grey.
Hold it like this
Yellow on top, white on the bottom, green facing you. That puts orange on your right and red on your left. Keep this grip for every step: the algorithms below assume it.
The centers never move relative to each other, so they always tell you where each color belongs.
Reading moves
Each face has a letter: R right, L left, U up, D down, F front, B back. A letter alone is a quarter turn clockwise, as you'd see it looking straight at that face.
An apostrophe (′, said “prime”) turns the other way, and a 2 turns twice. Lowercase letters turn two layers at once, M, E and S turn a middle layer, and x, y and z turn the whole cube.
Part oneLayer by layer
Seven steps and three short algorithms to remember. Slow, but it always works.
1The white cross
Find the four white edges: pieces with white and one other color. With yellow on top, bring them up around the yellow center, white facing up. That's the daisy. There's no algorithm here, just turning faces until each white edge is up.
Then, one edge at a time: turn the top until the edge's other color sits above its matching center, and turn that face twice. The edge drops to the bottom, white side down.
2The white corners
Find a corner with white on it in the top layer. Turn the top until it sits above its place, between the two centers that match its other colors, and hold the cube so it's at the front-right.
Repeat R U R′ U′ until it drops in with white facing down: once if white faces right, three times if it faces up, five times if it faces you.
A white corner stuck in the bottom layer in the wrong place? Hold it at the front-right and do R U R′ U′ once to bring it up.
3The middle layer
Look for an edge in the top layer with no yellow on it. Turn the top until its front color matches the center below it: an upside-down T.
If its top color matches the center on the right, the edge goes right. If it matches the left, it goes left.
An edge already in the middle layer but wrong? Put any top edge into its place to push it out, then put it in properly.
4The yellow cross
Now look only at the yellow edges on top. You'll see a dot, an L, a line, or already a cross.
Hold the line left to right, or the L at the back-left, and do F R U R′ U′ F′. A dot becomes an L, an L becomes a line, and a line becomes the cross.
5The yellow face
Count the corners with yellow facing up. With exactly one, hold it at the front-left and do R U R′ U R U2 R′, the Sune. If that doesn't finish it, do it once more.
With none, turn the top until the front-left corner shows its yellow on the left side. With two, until it shows yellow facing you. One Sune then leaves exactly one corner up.
6The yellow corners
Find two corners on one side whose side colors match: headlights. Hold them at the back and do R′ F R′ B2 R F′ R′ B2 R2, then turn the top until the corners line up with their centers.
No headlights anywhere? Do it once from any side, and you'll have them.
7The yellow edges
When one side of the top layer is complete, hold it at the back and do R U′ R U R U R U′ R′ U′ R2. Not solved yet? Do it once more.
No complete side? Do it once from any side, and you'll have one.
That's a solved cube.
Part twoToward speed
The beginner method takes about 120 moves. Speedcubers use CFOP: Cross, F2L, OLL, PLL. About 55 moves, with no pauses between them.
The cross, on the bottom
Skip the daisy and build the cross straight on the bottom. Use the fifteen seconds of inspection to plan all of it before you turn. Most crosses take six to eight moves.
F2L: the first two layers
Instead of a corner, then an edge, solve each white corner together with its middle-layer edge: join them in the top layer, then put the pair in with one motion. These four come up most. The corner is always above its slot at the front-right.
There are 41 F2L cases, and most are solved by understanding, not memory. While one pair goes in, look for the next.
OLL in two looks
Turn the top yellow in two steps. First the cross: the line left to right, or the L at the back-left.
Then the corners: seven cases, one algorithm each.
PLL in two looks
Then put the top layer in place, corners first. Headlights on one side: hold them on the left for the T-perm. No headlights at all: the Y-perm.
Then the edges. Three edges cycling one way or the other, or two pairs swapping.
From here
Learn the full sets one case at a time: 57 for OLL, 21 for PLL, starting with the ones you meet most. Turn with your fingers rather than your wrists. And look ahead: while one pair goes in, find the next.
Then time yourself. The cube at the top keeps every solve, move by move.