Hi Rick, thanks for this great question!
String muting is indeed very important, particularly while using high gain tones.
The first step is to slow down while practicing and be very mindful of how your picking hand is positioned.
The most common way to mute the lower strings while playing on the higher ones is to lightly rest the side of your picking hand on top of the strings right next to the bridge. Now you can adjust how low or high your picking hand is positioned in order to change strings and still mute below the strings you are fretting and picking. Zoning in on how your picking hand moves while you play will allow you to analyze it and make any necessary adjustments to keep the unused strings muted.
As for the higher strings, it is more likely you will be muting them with your fretting hand. Say for example you are playing single notes on the G string, your fretting fingers will naturally curl around the higher strings. Again, zone in and analyze how your hand naturally sits and, if necessary, adjust the curl of your fingers to lightly touch the higher unused strings.
The more and more you practice these little adjustments and hand positioning, over time you'll find it starts to become second nature, and you won't have to think about it very much anymore.
Another valuable tool to see if it's working is to record yourself playing something you've been working on -- and listen for any extraneous noises from the unused strings. If you can hear something that's not supposed to be there, then go back, focus into your hand positioning, analyze and adjust. If it sounds very clean -- what you are doing is working!
Hope this makes sense and gives you something to go on -- please check in and let me know how it's going!
Mike
Keep rockin!
Mike Olekshy
GT Guitar Coach