I need advice about how to make an audio frequency analyzer
This really depends on how fancy you want to get. A "good" analyzer will typically do the following things.


Split the input into frames. The frames typically overlap and are windowed. Good choices are an overlap of 50% and a Hanning window.
Do an FFT
Select the center frequencies of bands. For audio good choices are octaves or third octaves (see http://www.engineeringtoolbox.com/octave-bands-frequency-limits-d_1602.html)
For each integrate the FFT energy around the center frequency. There are different choices for the integration windows. Good choices are trapezoids or butterworth windows. A really good window choice will make sure that all the energy is accounted for, i.e. that the sum of the band energies equals the sum of all FFT bins.


If the number of frequency bands is relatively small, it is more efficient and easier to simply use a bank of parallel bandpass filters and an RMS detector. This also allows to dial in the temporal behavior of the display nicely.