I find the best way is the "broomstick method". This is easiest done with two people but possible with one.
Catch up the goose, take it somewhere the others can't see (or you'll never catch the others: they aren't stupid!)turn it upside down holding the legs and supporting the weight with one hand on the breast. Have the legs and belly towards your body. Lower the bird so the head and top of neck are on the ground with the chin,as it were, on the ground and put a broomstick or similar over the neck just behind the skull. Tread down on one end as you do this to keep it in place and then stand on the other once the neck is in position, so you have one foot either side of the goose's head. Then swiftly and forcefully pull upwards. You should feel the neckbone give quite distinctly and it is unlikely you'll pull the head off! It'll flap a fair bit, but should bleed into the neck cavity. To be sure, though, I usually cut the throat right down to the bone and let it bleed freely. Messy, but 100% sure. Another method is simply hold the goose down, hit it hard on the head with the back of an axe, turn the axe over and lop the head off. I don't like that method much myself though.
Plucking: best to get as much out as you can while it is still warm. I have some great "rosin" stuff from France which pulls them all out in clumps which is a godsend as geese have so MANY feathers and tonnes of down and it takes forever! Finish off by running a blowtorch over the skin quickly and carefully to get rid of hairs and any remaining bits of down and then wiping over with an oiled cloth for a nice sheen and to remove the soot. I hang mine for around a week at Christmas: I think a little less this time of year as it is much warmer. Goes without saying, leave the innards in until you're ready to use or freeze.
Good luck!