Noise in Analog Read Serial from an instrumentation amplifier(ina122p)
&quot;noisy most of the time&quot; - good news: if it's not all the time, that's a sign it can be fixed.
&quot;bandpass filter is also not really helping&quot; - INA122P circuit likely has something wrong.  check any resistors involved. Edit: if this was about U2.2, see below as that doesn't look like a band pass...
&quot;regular solid core wire and alligator clips to connect to the electrodes&quot; -
basically 0 shielding with mini-rabbit ears at each end and a big loop area between the wires.
Eventually you will want something better than gator clips.
My main recommendation is that as far as wire goes, if you form twisted pairs they tend to be pretty darn good at rejecting noise at least for that part of the wire.  You will want to twist a signal along with its reference (such as a +V and a ground).
Some explanation of the circuit:
U2.1 is creating a 2.5V source that it uses as the reference 'ground' for U1.  Reason for this is to treat 0 as &quot;-2.5&quot; and 5 as &quot;2.5&quot; as op amp math is generally done on a +V, -V basis.  (Personally, I prefer to use voltage regulators to make power buses instead of this. As a result, I give a stink eye to circuits that use the op amp method and claim something about reducing part count.)
U2.2 is a capacitor shy of an active band pass.  It's a high pass filter that is helping preserve and even promote some of the high frequency noise you are picking up. So yes, you may wish to modify or even ditch it until you have high frequency noise under control.
If possible, you may also wish to look into operating at a higher voltage.  This can help improve your signal to noise ratio.  If your signal is essentially +/-2.5V, 250mV of noise would be 10:1, but if you can run at +/-5 V, that same 250mV noise contribution gets dwarfed 20:1.