If you want to look at the shape of the curve, you should sample at least 20 times faster than the frequency of the signal. This allows you to capture a sine wave, square wave, or triangle wave and still have it presented reasonably close to the real curve.

If you do know that the signal is a 100% pure sine wave then you can manage to sample just above 2*f and then recreate the signal. Getting too close to the 2*f limit means your samples can represent a very low-frequency aliased sine wave where it takes a very long time until you do get a sample at a true maxima. So even when you just recreate a sine wave from samples, it helps greatly if your sampling rate is reasonably much higher than the theoretical limit. And remember that sampling at an exact multiple of the input frequency may result in the samples always happen on the same positions on the input curve - so sampling at 4*n could get you two identical high values, followed by two identical low values - it's possible to recreate the curve but the number of samples used will affect the quality. More points into a least-square equation helps.

For a periodic signal possible to get a good synchronization signal from, it is possible to perform many captures of one period but with a varying offset on the samples - this is what is done by many oscilloscopes that have a high input bandwidth but not fast enough ADC to capture at 20 times the input frequency. The concept works well for some problems but is totally useless for capturing single-shot events.

But let's assume 20*f as sample rate. That means 60 samples/second for a 3Hz signal. 200 samples/second for 10Hz or 2000 samples/second for 100Hz.

So what is the bandwidth of your input signal? What base frequency and what amount of higher-frequency overtones you need to catch? That is what decides if your ADC will be fast enough. And that is what will decide if you need more than 19200 baud to transfer your measurements to the PC.

But connect-the-dots in the PC only works when the measurements are close enough in relation to the frequencies of the sampled signal.