Like the Animal
The bunny hop is essential for clearing obstacles on a fast track....or as a climbing starter...It is really the same move as for the backhop, but moving along at any speed.
For these silly signs on the road
Make it simple, start above some paper box, at a medium to low speed.
1° Push on the ground, lifting the front wheel and extending the legs. 2° Pull your knees upwards and push forward on the bars, lifting the back wheel. 3° Once the obstacle passed, release the knees, letting the back wheel touch down.

How to lift the bike in a bunny hop or back hop, or side hop?
Lifting the handlebars is the only trick, because your feet can't lift the bike. It looks like the feet are stuck to the pedals, but in fact they are just limiting the ascension of the bike. The higher you lift the handlebars, and the faster you push them forwards afterwards, with a forward twist of the wrists to force the whole frame to follow, the higher you can get the back wheel off the ground. Of course you need some synchronisation, this is the difficult bit that needs hell of practice.
You can practice with full brakes on (the safest way to understand the move). That leads you to the back hop.

For the impulsion to jump, think about it like jumping on your feet without a bike. You need to flex the knees, then full extension. On the bike, same business but with synchronised pull on the bars upwards to lift the front wheel while still in extension. Once you are fully extended, it's time to lift your feet, like you do to jump over anything. At the same time that you are lifting the feet, you need to twist the handlebars forward (its the only way the frame can follow, the entire bike rotates forward in a vertical plane). All the move takes less than a second.

Why aren't clip pedals recommended for this?
Because Its way too dangerous to have your feet stuck when you need to jump off the bike at any time for a safe crash. You'd rather land on your feet than on your back, believe me.
It helps a lot to have lets say a paper box to practice over, and build up the heigth progressively.

There are loads of useless variations on the bunny hop, like the "one footer" or "no footer", or 180°.
The next step is to land your back wheel on an obstacle, and stay there. It 's a variation of the bunny hop where you do exactly the same moves, but with much less speed, so that you land on the obstacle instead of passing it....

see that in the climbing zone