The most basic style of playing guitar is strumming. You just need to make the chord shapes with your left hand on the fingerboard, pressing the strings down firmly and changing chords smoothly, then strum the strings near the sound hole with your right hand.
Listen to the song that you want to play (a recording, performance or just your own version- to "hear" the rhythm clearly.
To start with, just strum down with your thumb, from the thickest string (actually the lowest in pitch) to the thinnest string. Do one strum for each beat. Try to make the first beat in each bar a bit stronger (e.g. ONE,2,3,4, ONE,2,3,4, etc, or ONE,2,3, ONE, 2,3 etc if the song has a 3/4 waltz rhythm ).
If you would like to watch a video clip of this simple strum, I have made a couple of short clips (very basic- horrible singing, sorry). They are:
Mockingbird- http://www.youtube.com/watch?v=0OH81RpPJxQ and
Tom Dooley- http://www.youtube.com/watch?v=czIUFFuPmTc
Practise until your down beat becomes smooth. Then you can try a little flick upwards between each downward strum, so that you can get a bit of rhythm going.
Once you are more confident, you can vary the pressure a bit more to mark the beat (DOWN up, DOWN up, if it's a song in regular 4/4 timing, or DOWN, down-up, down-up for a 3/4 song such as The Streets of Laredo.
Then you can try using your fingernails as well- perhaps just your thumb and first fingernail at first. Work out what's most comfortable. If you curl your fingers up so your first 2 fingers are lightly touching your thumb, with your wrist nice and loose, you can flick up and down smoothly, catching the strings on the up- and down-strokes.
If you really want to use a pick, you'll need to work out how tightly to hold it, and the best angle to use. If your guitar doesn't have a scratch-plate, I wouldn't recommend a pick. It's easier to pluck the strings with your fingers, too.
Once you've got the hang of the chord changes, you might want to add some right-hand patterns or arpeggios- plucking the strings in sequence from the bass* note (the string shown on each chord diagram as the heavy line) up to the top string and back down again. Here's an example:
Pattern 1 (for 2/4 or 3/4 time): pluck B*,3,2,1, 4, 3,2,1, B,3,2,1, 4, 3,2,1 (the rhythm is one- a-and-a , two- a-and-a, three -a-and-a, four-a-and-a, ...)
Practise until you can play this pattern very smoothly, for a nice rippling effect. Use your thumb to pluck the bass and 4th strings, and your first 3 fingers for the top strings (1 per string).
Pattern 2 (for 3/4 or 6/8 time) "Ripple Pattern": B,3,2,1,2,3; 4,3,2,1,2,3 (count one and-2-and-3-and; one and-2-and-3-and..)
* The bass note will vary with the chord- for E, it will be the lowest E note (i.e. the open 6th string); for C, it will be the lowest C note (i.e. the 5th string, which will be held down on the 3rd fret): for A, it will be the 5th string again but it's open this time; for D, it will be the 4th string, so you may need to vary this.
April, 2012: I've been adding suggested right hand patterns to some of my later song sheets, and have also written out my favourite pattern, which I use for Sounds of Silence (as well as many other songs), in the Updates, News blog.