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).
The Mandola is ideal for playing chords, but has a short enough neck to allow the playing of melody as well.
To a large degree, the mandola is played like a mandolin. The main difference is that it's larger and the distance between frets is longer, making it more of a stretch. For playing melodies, this gives the opportunity to adopt a different fingering pattern prefered by some. The same applies to the tenor banjo and bouzouki - anything with a longer scale length. Notice that the angle at which it is held becomes more upright as the scale length increases.
See How to play a Mandolin for the basics. Holding it and tuning it are similar. The left hand technique and right hand technique apply also, but there are options as mentioned above. As well as the standard left hand fingering, longer scale length instruments can be played in the first position by the following change: first finger plays notes on frets 1 and 2, but then, middle finger plays fret 3, index plays fret 4 and little finger plays fret 5. To play higher frets, second position becomes more important.
The fingering positions as they are known, relate to positions on a violin (or violin family i.e. viola, cello) fingerboard but are useful on fretted instruments too. The idea is that when you cannot obtain a note in first position, move the first finger up to the first fret that was played by the second finger and gain that extra space on the fretboard. Violin players will learn all the fingerboard notes at all (at least) 6 positions and can often play the same piece of music in different positions.
What is important is that moving up to second position means that first finger now plays frets 3 and 4, middle finger plays fret 5, ring finger plays fret 6 and little finger plays fret 7.
(On the mandolin, second position would be first finger frets 3 and 4, middle finger frets 5 and 6, ring finger frets 7 and 8 and little finger can fret 9 and 10. This can also be achieved on the mandola if your fingers are long enough)
Third position is similarly moving the first finger to the position normally played by the third finger etc. This is how one achieves higher notes, and it is worth practising so that you are not limited to tunes in the first position (although a very many are)