The FFT can help you figure out where the frequency is, but it can't tell you exactly what the frequency is. Each point in the FFT is a "bin" of frequencies, so if there's a peak in your FFT, all you know is that the frequency you want is somewhere within that bin, or range of frequencies.
If you want it really accurate, you need a long FFT with a high resolution and lots of bins (= lots of memory and lots of computation). You can also guess the true peak from a low-resolution FFT using quadratic interpolation on the log-scaled spectrum, which works surprisingly well.
If computational cost is most important, you can try to get the signal into a form in which you can count zero crossings, and then the more you count, the more accurate your measurement.
None of these will work if the fundamental is missing, though. :)
I've outlined a few different algorithms here, and the interpolated FFT is usually the most accurate (though this only works when the fundamental is the strongest harmonic - otherwise you need to be smarter about finding it), with zero-crossings a close second (though this only works for waveforms with one crossing per cycle). Neither of these conditions is typical.
Keep in mind that the partials above the fundamental frequency are not perfect harmonics in many instruments, like piano or guitar. Each partial is actually a little bit out of tune, or inharmonic. So the higher-frequency peaks in the FFT will not be exactly on the integer multiples of the fundamental, and the wave shape will change slightly from one cycle to the next, which throws off autocorrelation.
To get a really accurate frequency reading, I'd say to use the autocorrelation to guess the fundamental, then find the true peak using quadratic interpolation. (You can do the autocorrelation in the frequency domain to save CPU cycles.) There are a lot of gotchas, and the right method to use really depends on your application.
