First Four Faces (FFF)
If you can solve a Rubik's cube, you can probably solve everything but
the last layer of Megaminx. This is because the Megaminx's the corners
and edges can be worked with in the same way as the corners and edges
of a cube. Therefore it's not really a challenge to do this step, and
this step can be completed in lots of different ways. However, the
Megaminx is such a large puzzle that it's important to stick to a
system when solving up to the last layer.
While I started out with something completely different, I've come to
prefer Grant Tregay's "First Four Faces" approach. A "Face" in this
sense actually refers to a face of the Megaminx and the five edges
above/below it, such that it resembles the F2L of a cube. In the
picture below, the part of the megaminx above the red line is the white
face. Notice that this includes five edges that have no white tile.

I solve each of these faces in a very Fridrich-like way. I make a star
(same a cross on a cube), then fill in the corner/edge pairs.The first
face can be anywhere. The second should share an edge with the first,
and the third and fourth should each share an edge with both of first
two, so that when the FFF are complete, there are only two completely
unsolved layers - the last layer and another layer adjacent to it.
A word about color selection
The single biggest challenge of doing the Megaminx quickly is
recognition. Due to the size of the puzzle, it's just difficult to find
the right pieces quickly. However, it can be made a bit easier.
While on a standard Rubik's cube it is to one's advantage to be
color-neutral because this allows the solver to pick easier cases in
the first step, and the ease of finding pieces on a cube is not a major
problem. On the Megaminx this is completely different. Any time saved
by picking the best case will very likely be offset by the longer time
it takes to find the easy case in the first place.
Therefore it is to one's advantage to solve the most eye-catching
colors first, and to always solve the colors in a single order. On a
Chinese Megaminx there are twelve colors. White is opposite gray, red
is opposite orange, blue is opposite light blue, yellow is opposite
florescent yellow, purple is opposite pink, and slightly darker green
is opposite slightly lighter green. If one looks down on the white face
and looks around it clockwise, the colors next to it are, in order,
red, blue, yellow, purple, and slightly darker green.
White is the obvious choice to me to start with. After that my next
face is usually blue. I would've picked red, but red is next to
slightly darker green, which I'd like to avoid early on because I can't
tell it apart from slightly lighter green. After blue I do red and
yellow. After these colors are done as the FFF it leaves only the light
blue and gray faces, which are good choices for the end because they
don't stand out very well in the beginning.
But that's just what I do. Figure out what you like best, just make
sure to stick to it. I can see how someone might like
white-yellow-purple-blue and the FFF because that leaves florecent
yellow and gray for the end, which is a bit easier to tell apart.
Preparation - First Four Faces - Orienting Edges and Approach to Last Layer