I think josepp was targetting the discussion to people who don't already know what the fifths progression is--that's one of the reasons to teach the circle of fifths. But the answer to that is pretty simple: first learn the FCGDAEB sequence, for which you can create a mnemonic, like (off the top of my head) Fast Cars Grimly Drive Around Every Body. I'm sure someone has a better one.

The problem is that, in drawing and labelling a circle using only the "ABC" sequence everyone knows, you have to skip points on the circle, so starting from C, you can only label six points: 12, 2, 4, 6, 8 and 10. How do you label the points in between? A simpler solution than what josepp describes (that jumping across the circle bit) is just to remember that to the left of C you have to put F, then continue the ABC sequence.

But this raises the second problem: how do you know when to start adding accidentals (if you're the sort of neophyte who doesn't already understand all the intervalic relationships)? First learning the FCGDAEB mnemonic solves the problem. When you get to the end of the sequence forward, you start adding sharps (and if you continue on, double sharps); when you reach the end going backwards, you add flats. So starting from C and labelling the points in turn, you get:

  12 1  2  3  4  5
  C  G  D  A  E  B
  

(The numbers here just indicate the clock points; you wouldn't write them down.) Contining with sharps you get:

  12 1  2  3  4  5  6  7  8  9  10 11
  C  G  D  A  E  B
  B#              \ F# C# G# D# A# E#
  

Continuing backward from C we get:

  12 1  2  3  4  5  6  7  8  9  10 11
              Fb Cb Gb Db Ab Eb Bb
  C  G  D  A  E  B                \F 
  B#              \ F# C# G# D# A# E#
  

We now have, essentially, a spiral in the form of a circle, with all the single-accidental harmonic equivalents shown. Moreover, it's now clearer (following the jogs indicated by the slashes) which spelling you should use when figuring out the notes in the scale, the accidentals in the key signature, the relative major/minor and so on. This is equivalent to the first strip in my message above, but in circular form (and with potentially confusing overlaps). By continuing the spiral this way beyond what is normally shown in typical circle-of-fifths diagrams, not only are all sharps/flats of the scale/signature duly represented, but the diagram works for minor scales (and other modes) as well as major ones.

But let's focus on what's important: these representations aim to capture the usefulness of the fifths series: ...Bb FCGDAEB F#... It's good to know the (equal tempered) pitch equivalences, but that's something one picks up better from a semitone-oriented mapping rather than one organized in fifths. In practice, when moving about by fifths, you will rarely jump from the flats series to the sharps series, though once you modulate into five-accidentals territory, it may be easier to consider this a switch to the enharmonic key (for instance, Ab instead of G#).

So (you'll smugly observe) if you have to learn the fifths sequence to create the circle, why create the circle at all? Because it's a helpful visual aid when explaining things commonly related by fifths, like progressions, modulations, key sighatures, primary chords and relative modes. The fifths sequence, combined with the diatonic interval pattern derived from approximating the overtone series, could be considered the underpinning of our musical experience.

In presenting my ladder version, I mentioned transposition, which some have downplayed, saying that once you learn the patterns, transposition becomes a more organic matter of applying those patterns. I agree with that, except when you're trying to notate music. Then it helps a great deal to have an explicit note-to-note transposition map to reference.

Of course, many people these days use music-notation software which handles transpostion in a far easier manner. But if you're not so blessed, a tool like the strips can be a real boon. A pair of nested wheels would also answer the bill, as long as the wheel is large enough to contain the entire 35-note sequence (Fbb to Bx) without overlapping. In this case, a fifths organization is better than a semitone organization, because it just happens that the most common notes and chords--those corresponding to the diatonic scale--will be grouped together, whatever the key and mode.