There are many sources on Detection Theory. I think there are two relatively simple approaches to your problem. From the example graph I can clearly see a distinct signal. if the desired detected signal has the same trend/shape a good detection technique would be in the direction of matched filtering.
(1) Matched filtering is basically correlating the test signal with the measured signal. if the signals are very similar to each other, matched filtering would return a high value on which you could detect that with a threshold. A good starting point for signal detection would be this lecture slide.
(2) However, that might already be a bit too difficult since you mentioned to be new to signal processing. Another approach when I look at the signal is seeing that the noise is very low when the signal is not present. I'm not sure if the example signals have these distinct amplitudes when the signal is not present (yellow at ~15, purple at ~-730). But if this amplitude is know, we could simply check when the signal is different from this steady value. Or we could calculate how much the signal changes by calculating the gradient (or simply the difference between samples) for example. If the gradient is high, the signal would be increasing at a higher rate.
Maybe start by calculating the difference between samples, i.e.: $s[i]-s[i-1]$ for each sample. I highly encourage to just play with the signals when starting with signal processing. Try different functions on the samples.