It's a little easier to see harmonics on a spectrum plot than on a sonogram. That loses how things change with time, though, as you go through different notes. So I made a movie, using a screencap on the spectrum analyzer in Live, from a chromatic scale. Ugly, but I think good enough to get the point across. (Full screen HD makes it easier to see details.) Frequencies are along the bottom, and a spike shows the intensity in each frequency bin. The bins form a comb, with a spacing set by the number of samples in each spectrum. You can see the lowest peak at the frequency of the note being played, the first harmonic an octave up (a factor of two in frequency), the second harmonic at a 12th (a factor of three in frequency), etc. Each harmonic moves up together as the note changes to the next one on the scale, but they keep their relationship to each other. To me the most interesting thing, though, is how the amplitudes of the harmonics all fit inside a single overall envelope, which is basically constant. Live keeps track of the peak hit by any momentary spectrum as a dim line, and this line shows the curve of the envelope inside of which the spectra of all the notes lie. The envelope has a broad peak around 500-600 Hz, ie around C4. So you have this funny thing where for low notes, the fundamental is weak, there will be many harmonics, and the strongest harmonic might be the third or fourth, while for high notes, you'll have only a couple relevant harmonics, with the fundamental being strongest. So the character of the sound changes dramatically in different parts of the scale, but it all makes sense, because every note is ringing inside of the same envelope. This envelope is called a formant, which refers to a broad acoustical resonance. Formants are particularly important in identifying vowels, but also help define the timbre of instruments.
Fletcher and Rossing talk about bassoon formants, quoting 440-500 Hz for the lower strong one, and a weaker one at 1220-1280 Hz. That's not so far off what I see. Saying there's a shape doesn't tell you why the shape is there, however. Fletcher and Rossing also talk about a cutoff frequency, which seems to be some kind of low pass filter coming from the tone holes, and quote 300-600 Hz for the bassoon's cutoff frequency. Another source I have, a chapter in the Springer Handbook of Acoustics, quotes Benade as measuring a cutoff of 400-500 Hz for bassoons, and stating "specifying the cut-off frequency for a woodwind instrument is tantamount to describing almost the whole of its musical personality". So it's important, however exactly it works. Fletcher identifies the lower formant with "the transition at the tone-hole frequency cutoff", and the higher one with the reed somehow. Or maybe it's the cone angle, eg the French horn has a similar bore angle, and a similar sound to the bassoon, at least during sustained notes. Regardless of how exactly the formants are formed, it's clear that they have much to do with describing the nature of tone.
Added: Joe Wolfe's delightful acoustics site has more on formants.