hey i just started playing the guitar and i was wondering, am i susposed to mute the string(s) after i play a note or chord?

for example if i play 3rd fret 1st string then play 3rd fret 3rd string, do i play the 1st string 3rd fret, then release the 1st string to stop it from sounding, then play the 3rd string 3rd fret? or just hold my hand position and let both strings ring togetheras i play?

If i am susposed to release/mute after playing each note/chord how would i mute an open string such as playing open 1st string then 3rd fret 3rd string? i wont hav any fingers on the open string so how do i 'release' the string to mute it?

It would help me a lot if i could get some answers, thanks :)