I have been practicing muting for some days. However, I am not finding myself very comfortable with it. I use my right hand placed on the strings while the left had lightly placed over the fret. Is this the correct way to do it? Also, are there some exercises which I can follow to improve?
There are both palm-muting and muting.
Palm muting is what's normally used in metal punk and rock which gives a different sound. This is used by playing the notes/chords normally with your left hand why holding your right hands palm on the strings while picking. Try playing a chord until you find the right spot and right pressure for your hand.
Normal muting is done with your left hand and is used a lot in funk,jazz (and all other styles too) and gives a rhythmic feel to it. This is done by lifting your fretting hand a bit and just lightly touch the strings so they don't vibrate.
So you shouldn't need to adjust both hands for this!
It depends on what the goal of your muting is. My guess is that you're playing in a rock setting and feel you're too sloppy, because ringing notes are causing strange results through your distortion. I say this because when playing clean and especially playing acoustic, the ringing notes are part of the sound, and the heightened attack makes it clear which note is the melody note.
If this is the case, the question isn't really about muting, it's about playing clean and fast. The muting of the previous notes gets to be a side effect of getting to the next note. Crank up the gain so you can hear where your problem areas are, take it slow and get it clean, then work it up closer to production speed. Getting your technique together with that one exercise, whatever it may be, will get your picking hand, fretting hand and ears on the same page when it comes to more general playing.
If I misunderstand, explain more specifically what you're trying to do and we can give more specific suggestions.
Muting could be tricky to teach as it depends on applying just the right amount of pressure in just the right place. It's something that I've managed to improve by just doing it a lot.
For the right (picking) hand I rest the heel of my hand just in front of the bridge. For the fretting hand I just release the pressure a little.
Try playing the same chord or riff with and without muting until you find the optimal pressure.