Right or wrong, this is the way I always cut up a chicken or a turkey. It gets all the meat off and the cuts look pretty.
Start off with the chicken beast-side up and gently cut the skin between the breast and the thigh. You don't want to move the skin much so it stays on the meat. Pull the thigh away from the chicken and cut the hip joint. You can use a sharp knife or some kitchen shears.
Flip the chicken over and finish cutting off the thigh.
Hopefully you have something pretty like this.
Repeat for the other thigh.
Cut the wings off in a similar fashion, separating the joints and not disturbing the skin or muscle too much.
There is a piece of meat reserved only for the person who worked so hard on this beautiful bird!! On the backside, in the middle, there are two bite-sized pieces of meat. They are the most tender and soak up all the flavor! Yum!
Cut the breast off by starting with a very sharp knife just to the side of the breast bone. Gently saw through the skin and scrape/cut the meat off the bone.
Repeat on the other side. Try to get as close to the bone as you can so you get all the meat off and don't have to make a second cut.
Then you are done! One of my breasts didn't make it with it's skin but you get the point!
Good luck! Don't be intimidated! I was the first few times I did it!