How do I walk on my front legs?

You will need to tap S and W, but very quickly. Then you will have to use the same buttons to maintain balance. The easiest way is to press S using your index finger, then press W with your middle finger and release S again. You can also use F for slow motion which will help you to understand how to best control the goat.

To enter manual mode, rapidly alternate between W and S. As has been mentioned, using different fingers for the two keys helps a lot.

Once you start the manual, a graph will appear. Use W and S to keep the colored line somewhere on the graph. At the top of the graph is the score you'll get if you complete the manual.

To "bank" your points, jump and land on your feet.

The "manual" challenges require a certain amount of manual points in the same combo. You can finish a manual and enter another one, as long as the combo stays running.

Do note that you can only manual so many times in a row before your combo starts to decay. You might need to do a jumping trick or something if you need to break up the manuals over many different attempts.