To some people, punting comes easily but for some others it seems immensely complicated. It is not actually very difficult & the following are a set of simple instructions. They assume you are right-handed (if left-handed then do the mirror image of course) and you are using a Cambridge style punt, here are some advice & tips on how to punt.
It is likely you will get your right arm wet, punt seat cushions may be dirty and it might rain so don't wear clothes that will not recover from this. Also wear shoes that will not slip much on wet wood (stiletto heels are not recommended!). However, it is still possible to get a smart garden-party or even a may-ball look in the summer by crafty compromise.
Stand on the flat platform on the back of the punt. Feet apart (for ease of balance) with the right foot a bit to the rear of the left. Hold the pole on your right with your right hand lower down the pole. The left hand will normally be palm up and the right hand palm down. Grip the pole in the obvious way with fingers wrapped around one way and the thumb round the other. Your hands should be about 30 to 70 cm apart.
If the punt is stationary, hold the pole almost vertical (but with the bottom end pointing slightly backwards) over the water on the right hand side of the punt. Let it side down at that angle into the water. When it reaches the bottom of the river push it down and back further along the length of the pole. Do this by climbing your hands up the pole (Push with both hands until your right hand gets too low then, without releasing your left one, release your right hand and grab the pole with it above where left hand is. Continue pushing with both hands until your left hand is too low and then move that up likewise.). Continue until your hands are near the top of the pole. At this stage you can optionally get a further push by bending your knees; I recommend that you use this extra push because at this stage the pole is closest to the horizontal so the force acts in the most efficient direction (i.e. pushing the punt along not pushing it upwards) so this is where you can get a good speed up.
If the punt is moving, point the bottom of the pole slightly further forwards (even to the extent of pointing it ahead) before letting it slide down so that the flow of water under the punt will have pushed it back to the right position by the time the pole hits the riverbed. (Otherwise at high speed, the pole may not even reach the riverbed before the flow has rotated it almost to the horizontal.)
When you have pushed the pole as far as you can, pull it up back out the water and rotate it back to the original nearly vertical position ready to start again. To pull it up, just feed it through your hands (With your right hand lowest, loosen the grip with the left hand (but leave it in place to guide the pole) and bring your hands together along the pole (since your lower hand is gripping the pole, the pole will move up). Now grip with the left hand, loosen the right hand grip and move the hands apart again (which also moves the pole up). Repeat until the pole is out of the water.). It is possible to throw the pole up instead feeding it up but I don't recommend it because it is easy to drop the pole on the heads of the passengers while doing that and they might get a bit annoyed.
A slight twist of the pole (done with both wrists) about its length before pulling it up is usually enough to free the pole if it is slightly stuck in the riverbed. If that fails you can try to bring the punt to a halt by brute force (hooking a foot inside the punt might be required to stop the pole pulling you off when doing this at speed) but the safest solution for a stuck pole is just to let go of it and paddle back (using the punt's emergency paddle) to get it at your leisure.
One more hint: Following the above instructions literally will make the punt not only go forwards but turn to the right because always pushing the pole on the right will swing the rear end of the punt the left (you will be pushing the pole not only backwards but also away from the punt). To prevent this, the pole, though it is on the right, must be pushed straight back not out to the right. This can be done either by leaning over the edge of the punt (it is not as risky as it sounds, only your arms need to be over the edge) so that you are pushing in line with the pole. Alternatively, just angle the pole slightly inwards before you drop it in and keep it close to the punt edge so that it slides under the punt and, though the top of the pole is still out to the right, it is pressing against the riverbed under where you are standing. If when first trying this you find the punt swinging out to the left and yourself precariously remaining with the pole out to the right, don't lean on the pole - remember that the pole & punt are moving apart & you do not want to fall inbetween. You can simply drag to the punt back to place by pulling with the hand that is lower on the pole and pushing with the other (pulling with both would simply lift the pole off the riverbed) while keeping the end of the pole on the riverbed (it feels as if you are trying to scrape a groove towards the punt in the riverbed). It is inelegant but it works and you can then try again. It will take a few attempts to find out exactly where to push to get a straight line and it does vary between punts.
The simplest way to steer is simply to leave the pole laying almost horizontally in the water behind the punt for a while between getting it off the riverbed and lifting it for the next push. Whilst it is there in the water it can be used as a normal rudder. I.e. you alternately push & steer using the same pole. (If you are not used to a boat rudder here are more detailed instructions: hold your right hand lower than the left and stationary (it acts as the rudder's pivot) and move your left hand (also gripping the pole) right to steer the punt left and left to steer the punt right. The reversal is because you are steering the back of the vessel not the front.)
The disadvantage of this is that you are not propelling the punt while steering so it is not the fastest method.
If you find that the punt always turns to the right when you are propelling it then you can make use of this to steer by switching between left-handed and right-handed punting frequently. Do the same amount in each handedness to go straight or do more of one that the other the turn.
Disadvantages include: it is heavy work quickly lifting the pole form one side of the punt to the other; you may drip water on the passengers who might moan about it; and it needs a degree of ambidexterity.
My favourite method is just to use the propulsion method above but at the 'hint', vary the position that you put the pole in the water to steer. Angle it slightly further under the punt to turn left or slightly further away from the punt to turn right. In this way you essentially get the steering for free in terms of physical effort in return for a little mental effort in working out where to place the pole. If it sometimes goes wrong, you can still use method 1 at the end of the push to correct it, and when it goes right you don't need to bother with the separate steering stage so you go faster.
I have not yet seen the Oxford style in person but hear that it involves the punter standing, more safely, in the punt rather than on the end platform and with end with the platform used as bow rather than stern. Both techniques have historical justification. Cambridge leisure punts evolved from wildfowl hunting punts in reedy waters where the maneuverability from rear steering was crucial whereas Oxford leisure punts evolved from canal maintenance punts where speed & stability on the straight was preferable.
Please don't blame me (& no liability accepted!) if you fall in the river, get splinters from the pole, catch something nasty from pathogens in the water, get a chill, get embarrassed, or are otherwise adversely affected by these instructions. (Frankly, those are normal risks from punting :-) ). I was not an expert punter by Cambridge standards when I originally wrote this article and I originally wrote it for friends who did not want to punt (particularly in the rain!) but who wanted to bluff that they had punted. However I have - being a non-intuitive mover but a dancer nevertheless and a physicist & engineer with a good visual imagination & a pedantically detailed way of explaining things - an ability to observe, work out & describe motions in detail and many people have found my punting description useful.
Here are some links that you might find useful: