deciphering the frequency response from the input data
Since the signal is most certainly real-valued, you can ignore the FFT values after $f_s/2$ (or sample 300 in your plot) since they will mirror the first ones. With a sampling frequency of 2500Hz, the highest frequency in your signal can be 1250Hz. If you have higher frequencies and don't filter them prior to sampling, you will get aliasing. Look up the sampling theorem for more information, it's quite important when working with sampled signals.

For your actual question I'm not sure what you want to achieve. With an FIR filter you can attenuate certain frequencies. If your transients are strictly not within your desired frequency range, that might work. If they are in the same frequencies as your actual signal, an FIR filter won't help. Is that supposed to be a transient in the first plot where the signal goes from being clean to jiggly?

If so, you could probably just use a lowpass filter which cuts off at 50Hz or similar.