This is not intended to be a comprehensive guide to playing but it will give you a start. There are many tutorials on-line and there are many great tutorial books geared to different styles of playing (Celtic/Irish, Country/Bluegrass, Jazz etc).
Playing the mandolin consists of 4 things:
Assuming a right-handed player (otherwise certain things are reversed such as strum with the left and fret or chord with the right).
Apart from that there are a few other things that are useful to learn, such as changing the strings but that's basically it. The first two are relatively simple. The second two can be practised for a lifetime.
Hold the instrument on your lap (don't hug it tight or you'll deaden the sound), or get a strap and adjust it so that your arm is horizontal from the elbow.
The left hand will either fret individual notes for playing melodies, or shapes for playing chords. Getting the fingers to do this in the beginning might seem unnatural and awkward, but persevere as it does get easier.
You need to learn where the notes are on the fretboard and where the chord shapes are. It is probably better to learn in small manageable chunks than attempt to learn the whole fretboard or whole set of chords in one go. This can be both daunting and not much fun!
For melody playing, there are two (possibly three) ways to go. The first is to read music on the stave, the second is to read tablature and the third is to play purely by ear.
Music on the stave. This is an additional learning challenge if you don't already read music but if you do, it's a big advantage.
Tablature (Tab) is an easy-to-read place-your-finger-here notation to help you play if you don't read music. The drawback is that it does not indicate how the notes should be played, or for how long so you need a good idea of how the tune sounds before play it yourself. Simply place the appropriate finger, on the string indicated at the fret number given. Note the lowest string (G) is at the bottom. To start playing this, fret the third string a 4 (and pluck) and then 6 (and pluck), then on the 2nd string, pluck it open, then again, then at fret 2 … If you look at the score above though, the symbols differ showing that some notes are played twice as fast as others (those joined together).
These two lines above (score and tab) are the same notes for the same tune. If you find a tablature that has the score as well, it helps to indicate the length of notes and any other indicators of style.
Find a pick (plectrum) that you are comfortable with (light to medium is normal on a mandolin, slightly heavier on larger instruments but it's a matter of preference - try several different grades). Hold the pick lightly but firmly resting on your first finger between the tip and first joint and clamp it lightly with your thumb (pointy end towards the strings!). Normally the pick is held horizontally and the hand is moved up or down from the wrist.
The right hand technique for melody playing is one of strumming the right course of strings either up or down. There are two main techniques - a single pluck to sound a note before playing the next, or a rapid tremolo to sustain the note, especially in slow tunes. It is entirely possible to mix the two in a piece of music, as is playing a chord in place of certain notes.
Playing chords, requires a different technique from playing melodies, as all the strings have to be struck at the same time (or as near as possible). Sometimes, depending on the type of chord, not all the strings are struck but let's not worry about that just yet.