At the moment I can step through the frequency band of interest and take samples to estimate relative power at various frequencies. But this is slow. Is it effective to center myself in the band of interest, make sure my sample rate is high, and do an FFT instead? I'm just wanting to efficiently get relative power levels for frequencies throughout a band.
As far as I understand your question, you want to estimate the PSD in a large bandwidth.
What you're currently doing is tuning to one part of that bandwidth, receive samples at a low sampling rate, calculate the power in that small bandwidth, retune, and repeat.
So, yes, using a larger sampling rate will obviously reduce the time you need to sense a large bandwidth: A larger sampling rate will give you more spectral information.
The DFT->square method is one classical spectrum estimator. So, yes, instead of tuning $N$ times to receive an $N$th of the overall bandwidth $B$ by sampling at a rate of $\frac BN$, you could just as well sample with a rate of $B$, and calculate $N$-point DFTs in the same time. Since sampling the same time at rate $B$ gives you $N$ times the samples of sampling at $\frac BN$, the information would be the same (Nyquist/Shannon).
You might want to look into other spectral estimators, too – but for now, I don't really see a big disadvantage in simply using the DFT with an appropriate Window – and given the fact that neither the analog nor the digital filters on the RTL dongle probably are overly fantastic, I think you can just stick with a Hann window and be fine.