To understand the moves, you will need to hold your Pyraminx as shown in the following images.
The letters below each image indicate that you turn the orange-coloured section through 120° in a clockwise direction. Where you see a ' after a letter, for example, T', that means you turn the orange section in an anti-clockwise direction.
Step 1 - Solve Corners and Connected Triangles
Twist each corner so that it is connected to a little triangle of the same colour. Once you have done this, you will not need to twist the corners again. Now, using only the turns shown above (ie not twisting corners), you should be able to get it so that, on each face, there are 3 corners of the same colour with little triangles of the same colour connected to them.
Step 2 - Solve Top Layer Edges
In the diagram, the pieces coloured grey are the edges of the top face. In this diagram, blue is used as the top face, you can choose which to make your top face. Slotting in the centres is relatively straightforward once you get used to it.
Study the following diagram. The blue-orange edge belongs in the location shown in pink.
You can place the edge there with the following turns,
R' D R
This position can be set up by turning D if needed. You turn it back where it belongs afterwards.
Suppose you had the following situation,
You can position the edge with the following algorithm,
L D' L'
When you are finished, the top layer is totally solved.
Step 3 - Solve The Remaining 3 Edges
Yes, with the top layer solved, there are a maximum of 3 misplaced or twisted edges separating you from solving the whole puzzle.
Hold the puzzle so that the remaining edges are on the T section of the puzzle. You will have something like this,
In this case, the 3 unsolved edges are a clockwise turn away from their correct positions. The following move rotates them in a clockwise direction without messing up the rest of the puzzle,
R T' R' T' R T' R'
In the following diagram, the 3 edges are one anti-clockwise turn away from their correct positions. The move shown will rotate each piece anti-clockwise without messing up the rest of the puzzle.
R T R' T R T R'
If you are lucky, this will be enough to solve the puzzle. If not, you will be left with 2 edges, which are in their correct positions, but are twisted. This scenario is shown in the diagram below and the algorithm shown will solve the puzzle from that position,
R L' R' L T' L T L'