Basically, use the right hand to mute the strings above the one you're plucking, and use the left hand to mute the strings below the one you're plucking.
For me it's exactly this. I rest my Right hand thumb on the string lower (in pitch) than the one I am plucking, using pickup for the E string. The side of my thumb rests on the lower strings. For example when I'm plucking the G string my thumb tip is on the D, and the side of my thumb lightly brushes the A and E.
For strings higher (in pitch) than what I am plucking, I use the meat of my left hand fingers - whichever ones are
not doing the fretting- to rest against the string. If fretting with my index finger than my left pinky and maybe others are lightly resting on the finger board, across the higher strings. If fretting with other fingers, my index finger lightly touches the higher strings.
It takes no pressure to stop a string from unwanted ringing. You just need contact.
I never practiced this muting directly itself. But what I
did practice were 2 things that involve the same skills: 1.) Controlling note duration and 2.) Muffling (sometimes also called muting, confusingly)
1.) controlling note duration can be practiced with a metronome and quarter notes. Just play beats 1 and 3, but
make sure you are playing rests on 2 & 4. In fact thinking explicitly of rests / silence as
something you play with just as much attention and awareness as a plucked note is extremely useful for tightening up your rhythm. One "trick" I use often is to stop all my notes exactly before the snare drum hits on 2 & 4 -"leave room for the snare"
2.) Muffling is using your left hand (sometimes the right) to deaden the vibration as you pluck so you get more of a percussive thump - but not deaden so much that the pitch becomes vague. Muffled note make great rhythmic embellishments and anticipations. Inventing lines that alternate between full note lengths and Muffled can go a long way to making your playing more dynamic and expressive.
I also try to adopt a "Mute by default" stance while playing. I keep both hands on the neck and unless I am deliberately plucking a note, I am holding the strings still. Controlled playing starts with enforcing silence, IMHO.