A Beginner's Method
to Solve The Cube
Basics
Solving the Cube
 Step 1: Bottom Edges
 Step 2: Bottom Corners
 Step 3: Middle Edges
 Step 4: Top Edges
 Step 5: Solve Edges
 Step 6: Position Corners
 Step 7: Solve Corners
 Done
Extras
Example
Here is a complete example for unscrambling a cube with the strategy and the moves explained under step 1 to 7.
In almost all cases the cube is oriented such that white is on the bottom. We write for example "orange in front" and mean that the entire cube was rotated such that the orange face is in the front (and white is on the bottom).
Scramble the cube
 We begin by scrambling the cube. Start with a solved cube, white on bottom and green in front. Do U Ri U2 Li Bi D Fi Ui Ri Ui. 
Step 1: Bottom Edges
Use elementary moves.
Pick white as the color on the bottom. Turn the entire cube and look for edge cubies with a white sticker.
 Focus on whitegreen edge cubie.
Orange in front.


Bluewhite edge:
Orangewhite edge:
Redwhite edge: 
We did not work very hard on scrambling the cube, so the first step happens to be rather simple. Only the redwhite edge requires some effort. First we move it into the middle layer (R). Then we move the target location with Di so that a simple F puts the cubie in the right position in the bottom layer with the white sticker down. Finish with D.
Step 2: Bottom Corners
Use the Ymove to swap corner cubies from the Ulayer into the Dlayer along a vertical edge. Choose between Y and its inverse to get the orientation right at the same time.
Look for corner cubies with a white sticker. In this example, there does not happen to be a single corner cubie in the easiest position, which is the top layer with white on the side. There are two in the top layer with white on top. These are the cases that require two Ymoves (or three if we only do Y in the standard orientation).

Whitebluered corner: 

Whiteorangeblue corner: 
The whitegreenorange corner now requires just another inverse Ymove, but let us demonstrate swapping a corner out of the bottom layer.

Whiteredgreen corner: 
With green in front, we are in the standard configuration after step 2. We could easily fix the final bottom corner, too, but we leave this corner unsolved because it is part of our manouvering space.
Step 3: Middle Edges
Use Y to cycle edge cubies from the Ulayer into the Mlayer. In preparation of the Ymove, turn D to move the remaining free bottom corner into the corner position that is affected by the Ymove.
Look for edge cubies without a yellow sticker, but ignore the greenorange cubie which is a part of the spare vertical column.

Orangeblue edge:
Bluered edge:
Redgreen edge: 
After putting the Dlayer into its original position and with green in front, we are in the standard position after step 3.
Step 4: Adjust Edges
Recall that the Ymove does two edge swaps, but we may need an odd number of swaps. Let's fix this right away. We leave solving all 5 edgies to Step 5, where the last 2 fall into place automatically because we adjusted the edges here. Our example is close to a worst case. You might prefer to first solve one or two edgies to simplify the swap count.
Count the number of swaps of the remaining unsolved edges. If the number of swaps is even, do nothing. If the number of swaps is odd, do one elementary Umove.

Counting swaps: Do one U quarterturn. This leads to a 5cycle representing 4 swaps, which is an even number. 
We could have done one Ui, which puts two edges in the right position and leaves one 3cycle for a total of 2 swaps. However, rather than having the yellowblue and yelloworange edges in the correct position but flipped, it sometimes helps to have such edges in the wrong position. Then the Ymove can move them into the correct position with a flip.
Step 5: Solve Edges
Use the Ymove to cycle edges around the upfrontright corner. Use U to prepare this move. Do not forget to undo this move with the exact inverse of your preparations, or the evennumberofswaps property is lost.
First, we do the upback and upleft edges, which require some Ulayer preparations.
 Green front.
Blueyellow edge:
Redyellow edge: 
The remaining 3 edges are in the position of a standard Ymove cycle. In our example, only one Ymove remains to be done.
 Green front.
Remaining three edges: 
Step 6: Position Corners
Use Ymoves to swap corners between the upper layer and the bottom layer. Since only corner cubies are to be moved, we do a Ymove, a Umove, the inverse of the Ymove, and the inverse Umove. This cycles 3 corner cubies without affecting anything else. For the five unsolved corner cubies arranged in the standard configuration as shown above, use Y in the "upsidedown" version.

Yellowgreenred corner: 
Two more cubies have to be positioned this way, the remaining two fall into place automatically.

Yellowredblue corner: (Could have solved yelloworangegreen instead.) 

Yelloworangegreen corner: 
Step 7: Solve Corners
Using a particular Ymove twice in sequence rotates corner cubies without moving them (because we swap corners back and forth once).
Consider the upsidedown Ymove. 2*FiRFRi rotates the upfrontright corner cubie by 120 degrees in the counterclockwise direction. Clean up by either following 2*FiRFRi with 2 times its inverse, or by doing a total of 6 FiRFRi's.
Combine these moves with Umoves to orient cubies in the upper layer. Recall that you may have to turn the entire cube to apply this strategy to the cubie at the bottom of the vertical column.
In the example, all we need is one clockwise and one counterclockwise turn of a cubie in the top layer.

Rotate cubies in top layer: 
We Are Done
 Isn't it beautiful. 