The beginner method to solving the classic 3 x 3 x 3 involves seven (7) steps, Eight (8) algorithms (two of which are mirror images of others, one which is a modification of another, but there is another mirror you can learn to save time).
The seven steps are:
1. Place and orient the edge pieces of the first layer. This should be done intuitively.
2. Place and orient the corner pieces of the first layer. This should also be done intuitively, but may require some practice to get it done easily.
3. Place and orient the edges of the second layer. This requires two algorithms, a master and a mirror image.
4. Place the corners of the third layer. This requires one algorithm.
5. Orient the corners of the third layer. This requires two algorithms, a master and a mirror image.
6. Place the edges of the third layer. This requires one algorithm, but the mirror image of the algorithm will be extremely useful, so you might as well learn that as well.
7. Orient the edges of the third layer. This requires two algorithms, but the two are somewhat related.
Algorithms:
3. Placing and orienting the edges of the second layer:
Align the edge piece you desire in the third layer with the side that matches the side facing you, like so:
Then use the algorithm:
Tl Ru Tr Rd Tr Cc Tl Cw
If the situation in the figure is reversed, like a mirror image, then perform the mirror image of that algorithm:
Tr Lu Tl Ld Tl Cw Tr Cc
Repeat the step for each layer two piece you see in the third layer. If you don't see any pieces at all in the third layer, you might have to move one into the third layer by using the algorithm to insert a junk piece in its place.
4. Placing the corners of the third layer:
By the time you get this far, all of the third layer corners will already be in the third layer. To place them in the correct positions, try to align as many as you can by paying attention to the three colors of the piece. If you see two adjacent green corners, line them up on the green side.
It is possible that all four corners will be in the correct position after you do this, but if not, use the algorithm:
Lu Tr Ld Cw Tl Cc Lu Tl Ld T2
to swap the two corners on the side facing you.
For example, if the green side is facing you, but the two corners you see are green, yellow, orange on the left and green, red, yellow on the right, yet the red side is on the left and the orange side is on the right, you will need this algorithm.
Don't forget to check the other side to see if it also needs to be adjusted.
5. Orienting the corners of the third layer:
If the color on the top side does not match the colors of the tops of the corners, use this algorithm to adjust them. The algorithm will reorient three corners and leave the fourth unaltered.
This algorithm rotates all of the corners clockwise, except the one in the near left:
Ru Tl Rd Tl Ru T2 Rd T2
This algorithm rotates all of the corners counter-clockwise, except the one in the near right:
Lu Tr Ld Tr Lu T2 Ld T2
If there are two or four corners that need to rotate, then you may repeat the same algorithm from different starting positions.
6. Placing the third layer edges:
This algorithm moves the edge on the top layer right to the left. You can solve immediately, if you start with this figure:
The piece on the left will move to the back, and the piece in the back will move to the right. The piece facing you will remain the same.
Mu Tl Md T2 Mu Tl Md
You don't absolutely need it, but it can save a lot of time if you can do the mirror image of that algorithm, which still keeps the piece facing you stationary, but moves the piece on the left to the right side:
Mu Tr Md T2 Mu Tr Md
Here is a figure of a cube that can be solved immediately using the mirror image algorithm:
7. Orienting the third layer edges:
There are two algorithms that you will need to do this, and they both share the sequence: Ml R2 M2(r) Rd T2 Ru M2(r) R2 Mr Ru T2, plus an additional one or three moves.
If your top looks like this:
(what looks like the letter "I")
Use:
Rd Ml R2 M2(r) Rd T2 Ru M2(r) R2 Mr Ru T2
If your top looks like this:
(what looks, to me, like an arrow)
Use:
Cw Ml R2 M2(r) Rd T2 Ru M2(r) R2 Mr Ru T2 Rd Cc
Once you have finished these seven steps, you have solved the cube, then you can practice. When you get bored, check out the intermediate method.