Columns First
also known as: Columns for 3x3x3, PCMS variant
This is something of a variant of Kenneth Gustavsson's PCMS method. I tried to optimize it a little bit to create a method that could be used with relatively few extra algorithms. Because it differs in a few key ways, I've decided to write it up.
The first step is to create four F2L pairs by themselves, without worrying about the cross or centers. Don't think of this as a Fridrich kind of problem, but rather as a blockbuilding kind of problem. In addition to what's free in a Fridrich solution, you can also use the M and S layers, which means moves such as r U r' are totally fine. I suggest trying to plan out 2 pairs in inspection. The SpeedSolving wiki page for Gustavsson's PCMS method has a good explanation of this step, and a few worked-out example solves.
In this step, you just have to orient the last layer corners, without affecting the four pairs. There is a lot of freedom for this, so it's easy to find algorithms for each of the 7 non-solved cases. I suggest choosing your favorite OLL algorithm for each one, noting that the edge orientation doesn't matter. Singce this is a fast algorithm step, you can lookahead to the next one.
This step is a little involved and tricky. You want to solve the centers, insert the four bottom edges, and make sure the top edges are oriented too. Almost all of this will be done using moves of the M, S, and U layers. I really recommend inserting three of the bottom edges (while fixing the centers), and then solving the last one while orienting the top layer. That last step can be easily done in one look - Doug Li provides Step 4: PLL