How would I go about analyzing this SDR-captured 300 MHz radio signal?
Good work getting a sample into audacity for visual inspection. If I understand correctly, your screenshot shows the raw waveform without demodulation (as opposed to baseband audio after FM/AM/USB/etc demodulation). Some folks mentioned PSK and FM modulations which are similar, so you could try setting your demodulator to FM and capturing a new "baseband" audio sample and putting that into your sound program. 

FFT analysis is quite helpful as well against an FM waveform. Depending on your receive software, I like Gnuradio - depending on your technical goals you may or may not find gnuradio palatable but I still recommend it if you will be spending some time doing this sort of work. You can do just about anything with signals in gnuradio. 

There's a wonderful signal identification wiki here (link). Most of the pictures shown on the wiki are snapshots of waterfall FFT analyzers, such as those found on GQRX, SDRSharp, gnu radio, etc. Waterfall analysis is hugely valuable because it lets you visualize a FFT over time and observe signal characteristics that are periodic or otherwise hard to see on a 1-d FFT.

In addition to listening to your signal with an FM demodulator, try an AM demodulator, and there's even simpler modulations, some being self-timed, like pulse code modulation. PCM won't have much of a sound but will be easy to pull apart visually in audacity. 

One thing that's easily spotted on your signal when viewed in audacity is the phase shift, where the wave crest/trough makes what looks like a reversal or a little wave on top of a wave which indicates a phase shift. This phase shift, when demodulated with an FM/Quadrature demodulator should result in an interesting audio output.