Muting is kind of an art in itself. Although used as a technique to get certain sounds in music, it's general purpose is to stop strings from producing unwanted noise, so it's going to help you stop the unwanted string-ringing.

You can mute with the right hand (palm muting) and also with the left hand by muting adjacent strings with the fingers holding the chord notes, and even with any left over fingers or thumbs. This is trickier than palm muting but I believe all these methods are covered in the tutorials you've viewed.

You just need to continue practicing. Muting is not just a blanket string silencer, it can be very selective in which strings you mute. And it's not easy concentrating on that extra thing while you are trying to learn chords and just getting started.

One little trick (and there is a GT tip of the week on this one) is to use a scrunchie (women's hair ponytail thingy)and slip it over the nut area. Adjust it so that it just deadens the ringing strings. Ultimately, though, you'll want to get the muting techniques down pat.