you haven't mentioned what guitar you are playing (acoustic, electric...) I play an acoustic, so I can talk only about it...
start with 12th fret... it's the easiest...
Just to mention... many people use their ring finger on their freting hand for playing harmonics... it's not a rule... you can surely get them with other fingers, but it tends to be easier with ring finger, especially for beginners... it surely was for me...
if nothing of the mentioned helps (freting hand: very light touch right over the fret, using ring finger, removing the finger immediately after picking a string; picking hand: picking harder with your nail) then the problem is probably in guitar's intonation... you can find that out, as darryls said, by trying to get the harmonic by moving finger a little above/below the 12th fret... if that is the case you can remember it and move the finger at that position whenever you're playing harmonics... but I would suggest a repair... the repair means your saddle should be adjusted (as I figured out Ultimate Garage Band, Guitar Tech forum, can help you)... bad intonation practicaly means all your frets are "displaced" (not in general case, though)... your guitar would never sound ok... you'll always be out of tune... that frustrates as you'll think that you suck, and not the instrument... you'll end up quitting... (that much of postive attitude
but I think that's a bare truth...) anyway let's think positive, and hope it's technique and not the intonation...
post about your progress... have this tips helped...
long live... bye...