How to eliminate the noise that is generated after the A\D conversion?
I would start using twisted pair where is possible, by twisting the pair you reduce the noise quite a lot.

And then, the simplest thing would be to make a high pass RC filter, where the cutoff frequency should be above 80HZ, the latter is to remove the electrical noise produced by the 110/220 power outlet (which uses 50 or 60HZ). I recommend you putting two of those filters in series, so you leave the signal to the output much cleaner, although with one you already solve a lot.

Also, you can look into this article, it's very instructive.