Calculating FFT for only part of full frequency range?
I was under the impression that the FFT was inherently calculated at every frequency 0->(Sampling Frequency)/2 distributed in bins of width Fs/(2*N). 


This is roughly correct. A discrete fourier transform (DFT) produces output for frequencies between \$-F_s/2\$ and \$F_s/2\$. If the input data is real-valued (rather than complex) the negative-frequency spectrum will just be the complex conjugate of the positive-frequency spectrum, so some time can be saved by not calculating the negative frequency bin values. 

The spacing of the bins is \$F_s/N\$, not \$F_s/(2N)\$.


  I'm not entirely sure how to go about performing the FFT on only a portion of the maximum frequency.


To calculate a small number of bins, there is a method called the Goertzel algorithm for calculating one bin at a time.

As a rule of thumb, it's typically more efficient to calculate \$M\$ bins of a DFT by the Goertzel algorithm when

$$M \le \frac{5 N_2}{6 N} \log_2(N_2)$$

where \$N_2\$ is the next power of two greater than \$N\$.