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How do I increase speed of my output axle 24 times using 8, 16, 32, and 64 tooth gears using LEGO Mindstorms components? I checked Sariel's website but it was no help.
You can't do this with the gears you've listed, but I'm not sure that you've listed the proper gears. I don’t believe that 32 or 64 tooth gears exist. I at least don't have any.
You can do this by combining common 8, 16, and 24 tooth gears. The following gives 3:1 × 2:1 × 2:1 × 2:1 = 24:1.
You cannot increase the output speed by 24 times using a classical gear system, and the gear teeth you specified, because 24 is not a product you can make from 2, 4, 8, 0.5, 0.25, 0.125.
You CAN, however make a CVT, which takes two input speeds and divides them to give an output.
jncration is correct.
Sometimes the gears on axles helps.
Required, but never shown