The method described on this page is called the Ortega method, which is slightly more advanced than the normal LBL (layer-by-layer) method you might have taught yourself. If you learn this, it will give you a surprising boost in speed over LBL. Why? Because your first layer is much easier, and your "PLL" is much faster than normal as well. The Ortega method is actually a "corners first" 3x3 method, but it adapts so well to the 2x2 that most people just know it as a 2x2 method. Here's how it works:
Step 1: Make a white side, but not a white layer. All you care about is that the 4 white stickers are together, it doesn't matter if the pieces are in the right spots relative to one another (In fact, it's better if they aren't!).
Step 2: Orient the top using your favorite OLL algorithms, since on the 2x2 there are no edges, you can adapt several 3x3 OLL algorithms to fit each 2x2 OLL case.
Step 3: Permute all the pieces at once! If you have to permute pieces in just one layer, you can use a normal PLL algorithm, but if you have to permute pieces in both layers, you get to use a (much faster) Ortega algorithm!
Which sort of 2x2 should you use? There are two main brands: Rubik's and Eastsheen. Neither one is fantastic, but Eastsheen is much better than Rubik's for a couple of reasons. Firstly, the Rubik's 2x2 is far too small to be speedsolved and the mechanism doesn't allow for corner cutting. Eastsheen doesn't really allow for corner cutting either, but the difference is that the Eastsheen 2x2 doesn't jam when you try to cut a corner. Also, the Eastsheen 2x2 is of a larger size which is suitable for speedsolving. One last difference is color scheme. Rubik's 2x2s bear the Japanese color scheme (yellow next to white) which can make recognition more difficult. Eastsheen has a standard color scheme, except that orange has been replaced by fuchsia, a strange pink/purple color. The permutation images on this page do not take this into account. I tried it out and they looked pretty nice, but it's easy to mistake fuchsia for red, so I changed them to the more conventional orange.
R U R' U R U2 R'
R' U' R U' R' U2 R
F (R U R' U') F'
(R U R' U') (R' F R F')
F R U' R' U' R U R' F'
R2 U2 R' U2 R2
y' R' F R2 U' R2' F R
Note: This algorithm only works for Ortega.
Interpreting the images: The left square represents the top layer as seen from above; the right square represents the bottom layer as seen from above.
R U2 R' U' R U2 L' U R' U' R
F R U' R' U' R U R' F' R U R' U' R' F R F'
You can probably now see why it's better to not to complete the first layer. Compare XLL cases with PLL cases... XLL algorithms are much shorter!
R2 U F2 U2 R2 U R2
z' R2 U2 R2
These last two are slightly more difficult to recognize. Looking at the front, the first case looks like a "7" and the second one looks like an "L".
R U' R F2 R' U R'
L D' L F2 L' D L'