Find the goat by checking hedges for hole, looking up in trees (yes, trees!), flower gardens, inside your house, down the road, over the river, etc. Goats naturally explore and can be very curious.
Approach slowly from an angle where the goat can see you. Goats know where their escape routes are, and they are faster and more agile than you, so you don't want to startle it.
Figure out what to grab when the goat starts running. If the goat wears a collar or other such device, try that. Horns can be used as handles as well, but are not always as strong as they appear and can break.
To pin against a wall, place one knee in front of the shoulder blade and the other knee behind the ribs and lean into it. Ideally the head should be held up as well.

A goat that has been caught.Once caught, the goat should be tethered to a secure object by a chain or strong rope.

Tips
When on a lead of any kind, do not stand in front of the goat's head or all brakes will go on, you generally have to push them.
Food always goes down well, but requires patience.
It is not good to "gang up" on a goat with a bunch of people, as this will cause panic. However, it may be good to have a partner or two that can help move the goat once it has been grabbed.

Warnings
If it rears, it is not good. If the goat rears taller than you are, it will aim to come down on your head. It generally becomes a game of chicken, and is one you want to avoid, especially if the goat has horns. If this happens, put your palm firmly to it's forehead, it should either come down safely or go over backwards.
Goat horns are sharp and serrated. Therefore, try not to get on the wrong end of them, as they will cause damage.
If the goat starts scrambling make sure your grip is strong and your hold is tough. You may be taken for a short ride, but typically, a strong hold will force the goat to do what you want it to do.
While unlikely, if the goat falls over, you should be wary of the legs. They are powerful and pointy