The guitar is a very physical instrument and its going to do all that. You have to mute with WHATEVER finger, palm, wrist, anything you got on EITHER hand to use that will mute unwanted string noise. I used to use my 2nd, 3rd and 4th finger on my picking hand to stop strings from ringing and the side of my thumb to stop the bass strings from ringing. Theres no easy way to do this. Theres no magic technique for this. Turn your amp up loud and start trying it and if you get noise figure out where its coming from and mute it with whatever you have available.