My strategy has changed. Now I solve all the corners with a few twists, then all the edges in no particular order. I still use the Move for the edges.
The little corner tips can rotate, but are always attached to the same pieces, so when I think of a corner piece it is the tip plus the piece it is attached to. There are three edge pieces surrounding each corner. A corner and its edges make up the small pyramid that, along with the base, makes up the whole puzzle. The base is comprised of all the pieces that have the same color. The base has three corners and three edges, one edge between each pair of corners. So if the base has 6 pieces and the small pyramid has 4, there are 10 pieces to the puzzle altogether. You can also see it as 4 corners and 6 edges. Each piece has a home spot. Every piece is home when the puzzle is solved. We call a piece the owner of its home. When the puzzle is scrambled, at least some of the pieces are visitors. That is, they are not in their own home spots, but in another piece's home spot.
In my cube ramblings I sometimes refer to online solutions that I have learned or at least looked at. I have not looked at Pyraminx solutions online. The best solution depends on what you are after. Want a strategy that requires the fewest moves? Want a strategy that is good for speed solving? Or do you want to simply figure out on your own how to solve the puzzle, and develop your own strategy for doing it consistently? That is where I am in my puzzling adventures now, so that is why I no longer look at how other people suggest solving a puzzle. Rarely ever. I have included the explanation of how to solve the Pyraminx below to document for myself how I do it. If you want to compare my method with yours for some reason, it is there for you to try to figure out. Or if you have tried to solve the Pyraminx and need help, then maybe this will help.
Solve One Layer
To solve the Pyraminx I like to start with one of the bases. First I solve three corners relative to one another. I think this can always be done in at most 3 twists. Then with those corners on the bottom I insert the edges into place. This usually takes 4 twists per edge. I'm going to call the small 4-piece pyramid on top, the top, as in the top layer. The 6-piece base is on the bottom. To get a base edge piece home that is on top, hold the puzzle so the home spot is in front and spin the top so the piece is in the back. There are two base corners in front, one on the left and one on the right. The left corner along with the three edges that surround it make up a small pyramid that I will call simply, the left. Guess what I mean by the right.
Now the edge piece that is in the top and at the back has two sides, left and right. If the sticker that belongs on the bottom of the puzzle is on the left, turn the left so the visitor comes to the top. Turn the top to replace the visitor with the owner. Turn the left to take the owner home. Guess what you do if the sticker was on the right.
Continue this process until the base is solved.
Solve the Top—Flipping Edges
Now twist the top to solve the final corner piece. It may be that the top edges are solved. It may be that they are in their correct positions but two of them need to flip. Or it may be that none of them are in the correct place. If two need to flip hold the puzzle so they are both on the bottom layer (the base), and so that one of them is in front. Let's call this piece, Piece 1, and the other edge that needs to flip, Piece 2. Turn the right so Piece 1 moves to the top. Turn the top so Piece 1 moves to the back. Turn the right the opposite direction as before. Turn the left so Piece 1's home moves to the top. Turn the top so Piece 1 comes to the front left. Turn the left so Piece 1 goes home oriented correctly. When an edge is flipped it is not oriented correctly. When it is oriented correctly, it is not flipped. Now turn the base so Piece 2 comes to the front. Turn the left so Piece 2 moves to the top. Turn the top so Piece 2 moves to the back. Turn the left back into place. Turn the right so Piece 2's home moves to the top. Turn the top so Piece 2 comes to the front right. Turn the right so Piece 2 goes to the bottom. Turn the bottom so the puzzle is solved.
Whoa! The above is a 14-move sequence. I like it because it makes sense to me. It is logical. I can follow the piece around the puzzle and see why it is flipping. I can use a similar technique on cubes to flip edges and twist corners. But there is another way to flip two edges that only takes 8 moves. Really it takes 9 if you count the roll of the whole puzzle in the middle. But still that is a big improvement movewise and there is something I like about it too. It uses The Move.
The Move start Right (The Move R) = R↓ L↓ R↑ L↑
Let R↓ mean turn the right counterclockwise, so that the edge on the front right goes to the front bottom.
Let L↓ mean turn the left clockwise, so that the edge on the front left goes to the front bottom.
Guess what R↑ and L↑ mean. Guess what The Move L would be.
Let Roll R mean Roll to the Right, which means roll the whole puzzle so the front spins clockwise 120˚.
OK. To flip the edges at the front left and front right positions you can do The Move R, Roll L, The Move L.
Solve the Top—Cycle 3 Edges
What do you do, though, if when you solved the top corner, you had 3 unsolved edges in the top?
Move one of the edges to the bottom and then around to the same side as the other two. (This is called a setup move.) Once all three edges are on the front they can be moved into their proper places using The Move! Just make sure when you move the piece to the bottom and around, that you do it in such a way that exactly two of the three pieces need to flip as they cycle. Roll the puzzle so the piece that does not need to flip is in the top layer and needs to move to the other top edge in front. If it is on the right do The Move L. If it is on the left do The Move R. Then do the two twists necessary to solve the cube. (That is, undo the setup move.)