The specifics of warp speed are never discussed on screen in the original series. We know that warp five is faster than warp four and that the Enterprise usually sticks to the lower end of the speedometer, even when time is short; consider Captain Kirk’s order from The Squire of Gothos: “Ahead warp factor three, Mister Sulu. Colony Beta Six wants their supplies. Let’s get across this void in a hurry.”

Other than that, we can’t make canonical statements about how fast each warp factor is. But we can cite the next-best thing: the TOS show bible and The Making of Star Trek, both written or approved by Gene Roddenberry. Those sources do specify speeds but they also contain what can only be a typo on warp factor three.

The show’s bible, third revision, has this to say about warp speed:

So warp factor one is the speed of light: 186,000 miles or 300,000 kilometres per second. Warp two is eight times the speed of light, and warp four is sixty-four times. That makes the formula speed = wf³ x c, where wf is warp factor and c is the speed of light. 

Warp	Multiple of c
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729

But that means the show bible got warp three wrong. It should be 27c, not 24. That error also appears in The Making of Star Trek, by Stephen E. Whitfield. On page 191, Whitfield had the following, copying his numbers and some of the language from the show bible. 

…greater problems result when it becomes necessary to express a speed many times faster than the speed of light. STAR TREK dialogue solves the problem by measuring all faster-than-light speeds in terms of “Warp Factors.” Warp Factor One is the speed of light. Warp Factor Three is 24 times the speed of light. Maximum safe cruising speed of the Enterprise is Warp Factor Six, or 216 times the speed of light. At Warp Factor Eight (512 times the speed of light) the ship’s structure begins to show considerable strain… Warp Factor Six is therefore exceeded only in instances of extreme emergency.  

The formula was changed for The Next Generation. The Star Trek Encyclopedia by Mike and Denise Okuda has a good explanation of the TNG math but, as this is a TOS blog, I won’t go into it here. 

It’s interesting to note that Fotonovel 11, The Deadly Years, states correctly that warp 8 is 512 times the speed of light. Those books were gems when they were published almost 50 years ago and still are today.

Postcript

While Kirk typically kept his ship to the slower warp speeds, Captain Pike had no such compunction. In The Cage, he casually ordered Tyler to take the Enterprise to warp seven and no one looked surprised.