I have experimental data where I collected data points over time and my "signal" looks oscillatory in nature (y-axis values are more or less constrained in a y-axis bandwidth), but the curve is irregular and not a pretty, or regular sinusoid so there is no visible repitition to visually determine the signal's frequency. All peaks and valleys of the oscillations appear to be the same thickness so my signal's oscillations never appear stretched out.
I collected my data points once every nanosecond to get this curve and I now want to do a Fourier transform to see what frequencies are in this signal. If I pick my sampling frequency for the Fourier transform to be once per nanosecond, just like the frequency at which I collected my raw experimental data, am I safe from having to worry about aliasing?
You need to know the bandwidth of the underlying signal you sampled to know if the measured data is aliased or not. If the signal is aliased, it already happened when you collected it at a finite sample rate (1 GHz in your case).
But maybe you're asking if you can decimate your signal to a lower sampling rate. Again, the answer is you don't know until you know the bandwidth of your signal. Assuming your 1 GHz sample rate was sufficient, you could take the FFT of the signal and measure the bandwidth. This will tell you if the signal is significantly oversampled and can be decimated, or not.