It sounds simple, but developing good technique for mounting and dismounting your bike will make you a safer and more efficient bike rider - and faster for those occasions when time is of the essence, like racing.
Mountain biking techniques - mounting & dismounting
To change gear before you set off, hold the bike's saddle and lift up the back wheel, and turn the pedals with one foot as you change gears
Before setting off, check your saddle is the right height and in the correct position (see also
Getting the right size bike). Also check the size of gear engaged - with a very large gear (big chainwheel, small sprockets) you will struggle to get going - if you are starting uphill you may not get going at all! Too low a gear and the pedals may spin around before you get any purchase with the tyres on the surface. If you need to change gear before you set off, hold the bike's saddle and lift up the back wheel, and turn the pedals with one foot as you change gear with your other hand.
Mounting technique
If you don't use either toeclips or clipless pedals the technique for getting on is simple, but both toeclips and clipless pedals require some practice, both for clipping in and out of the pedals. It is a good idea when you first get clipless pedals to practice using them both stationary - leaning against a wall - and then in use on a quiet road or track. Similarly, if you use toe-clips and straps, you must ensure that the straps are not too tight and that you can release your feet when you stop.
Both toeclips and clipless pedals require some practice, both for clipping in and out of the pedals
It is best to mount your bike from the left to avoid getting covered in oil from your chain, and for reasons of safety in the UK (oitherwise you could be standing in traffic!) Obviously, if you ride on the road abroad, you do still need to be able to mount your bike from the opposite kerbside.
Stand beside the bike with both hands holding the handlebars. Lift your right leg over the back of the saddle and place your right foot onto the pedal (flick the pedal round to engage the toe-clip or cleat mechanism). Bring the right pedal up to the two o'clock position. If you can, slide your backside onto the saddle. You may only be touching the floor with the ball of your left foot, but that's fine, it's only there for initial balance.
Look behind you to check that there's no approaching traffic. If all is clear now, then look ahead to where you want to steer, push down with your right foot to begin pedalling and push off with your left foot then bring your backside onto the saddle (if it's not already there). As the left pedal comes round the 'top dead centre' place your left foot on the pedal and continue pedalling. If you do not engage the toeclip or clip mechanism immediately, carry on pedalling for one or two more revolutions to gain some momentum before trying again. Steer a straight line, about 50-100cm from the kerb, or if you're off-road, select the best course.
Once you are competent with the above then you can then speed everything up - so that you almost jump onto the saddle and begin pedalling straight away - this is obviously faster and smoother and is the technique used by cyclo cross riders.
Dismounting technique
When riding off road you will inevitably come across obstacles that you cannot ride though (or over) and many rides will include gates and styles where you will have to dismount. The aim is to get off the bike smoothly and safely so that you can then continue on foot before remounting. Uneven and slippery surfaces can sometimes make this more of a problem.
Dismounting your bike is much easier if your bike is properly adjusted to fit you (See also
Getting the right size bike). In particular, if your saddle is too high this may cause problems getting off the bike. Check too, that your brakes are working well and that you are familiar with them (See
Braking technique).
Mountain biking
Begin to brake as you approach the obstacle; do so well in advance so that you do not have to panic. Use both brakes evenly, unless it is downhill or the slope is off camber, in which case use the front brake very lightly. If you are heading uphill, gravity will slow you, so you needn't worry too much about braking. (See also
Braking technique). Whether you are left or right handed it is always best to get off the left side of the bike. This avoids you getting covered in oil from the chain and gears!
Once stationary, place your left foot on the floor. You will probably need to lean the bike slightly to the left (if the bike is set up properly you will probably only be able to touch the ground with the ball of one foot when seated on the saddle). Move off the saddle to place your left foot flat on the ground and then take your right foot of the pedal and bring your right leg round and over the back of the saddle so that you are standing with the bike on your right.
Advanced
Although left-sided dismounts are always preferred, it is worth practicing some right-sided dismounts, just in case you get stuck somewhere where a left-side dismount is impractical (e.g. due to stinging nettles, or barbed wire on a narrow path). As you become more confident you can change the technique slightly to a faster one where you are dismounting while you are still travelling, so that you can get off the bike without even stopping at all!