I enjoy the whole process, too. Since I work full time, I usually have to do the majority of my cooking in the evenings. So I'll bake the chicken one night, let it rest, and then it gets to hang out in the refrigerator overnight. (After I eat the oysters, of course.) Then the next evening, I do what is so elegantly called, "picking the carcass." Generally I'll keep enough meat for whatever I'm creating for dinner that night, (last night happened to be a chicken pizza) put the rest of the meat back in the fridge, and the bones, skin, and other various bits gets thrown in the pot along with whatever extra veggies are sitting around.
This is where some people get rather incensed with my process. There are die hard you-must-do-it-this-way-and-none-other stock makers out in the world. I feel it's about economizing, so I tend not to buy extra ingredients for something that costs me nothing but extra time. This go-around, I had fresh garlic, some extra carrots and celery from a previous dinner, and a collection of herbs. One time, I had a bunch of cilantro that I threw in the pot. It ended up tasting really good, but the stock had a green cast to it.
The stock bubbled nicely throughout our dinner and after. Once it was down to a nice level and color, I strained it into a large bowl using my metal colander. Then I strained it with a funnel and small strainer into small jars that went directly into the freezer. Most people like to wait for it to cool and skim off some of the fat, but you can do that with the frozen version too, and sometimes that little bit of extra flavor really adds to the dish, so when the mood strikes, I end up using the whole jar, including the top surface.
Finishing this task always leaves me feeling warm, fuzzy, and accomplished. Wouldn't it be nice if everything was so fulfilling?