You can increase the signal to noise ratio by using a filter that has a higher gain in the portions of the spectrum where your signal has higher power, and lower gain where your signal has lower power. You do that by creating band-pass filters centered around the parts of the spectrum where your signal has the highest power level. The result is that you throw out the noise across most of the spectrum. You also throw out some of your signal, but only in the parts of the spectrum where it had much lower power levels.
From your FFT graph, your signal has the highest power levels at 150Hz and 200Hz. There is also a little more power at 50Hz and 225Hz. If you take your FFT data array and zero out all the samples from 10Hz to 40Hz, 70Hz to 120Hz, 230Hz and onward, and then take the inverse FFT you will get your original signal, with some minor distortion, and most of the noise removed.
If you are using MEMS accelerometers, the RMS noise is often proportional to the square root of the sample rate. So sampling at a lower rate can often lower the noise power.
If the noise were occurring at some particular frequency you could just create a notch filter at that frequency. Or even simpler, take the FFT of your results, set the values in the FFT data array at the noise frequency to 0, and then take the inverse FFT to get your original signal minus noise.
The problem is that your FFT graph shows the noise amplitude as pretty flat across the in the frequency domain. So it would be classified as white noise. Therefore you can't use a notch filter to remove the noise.
If you just want to detect what operating mode some machine is in based on the vibration (and you don't really care about the shape of the signal itself), then you can just put a band pass filter at the frequency you know the vibration to be occurring, and then trigger based on the amplitude going over a certain threshold.