There are several ways to compost at home. 1) Regular pile compost (hot or cold) 2) Dig-a-hole Compost 3) Vermicompost. OK, I named the second one myself, but it is basically as described. If you have access to ground (a small yard, somewhere with soil) you can perform the first type types. If you live in an apartment and have no yard go straight to the Vermicompost description below. To Do #1 or #2 is very simple, all you need are 4 things - GREENS, BROWNS, AIR, WATER. Greens include any vegetable scraps, green leaves, etc. Browns include naturally fallen leaves, paper, etc. You basically want a 1:1 ratio on Green:Brown. Mix it, put it outside on some soil, and water it once in a while. I threw some worms on my pile to start and I don't even have to turn the pile. The pile smells great, like the soil after a rainfall. If you want to turn the pile once in a while to introduce Air, it helps decomposition, but you don't have to. If you don't have room and want to hide the compost pile, just dig a hole and drop it all in there. I do something in between, I dig a shallow hole, mix the veggie scraps with the soil, and put brown leaves on top (see picture) to conceal the food. It works wonders. Soft items are gone in a few days. Banana peels disappear after 1 week. Corn cobs take several weeks and require me to break it down after it dries.
If you're indoors you can do #3) Vermicomposting, which is composting with worms. I do this at home as well and you can see my bin in the picture. It's a bin from Wriggly Wranch and I have decorated it. I keep this bin inside during the winter to keep the worms alive. There are other creatures working for you besides the worms, but you mainly only see the worms unless you observe closely. This one is also very easy and fun, and you only need one ingredient: GREENS for this method. I bury about 1 cup of food scraps every few days and the worms eat it up. I don't feed my worms tough to eat stuff like corn cobs or avocado skin. Nor do they like too much citrus. The composting speed is slower than the outside pile. But it works all through the winter.
I hope you will explore home composting, it's fun, educational and green!