We don't peel or core or anything, just cut in quarters. I'll claim it's so that we get the nutrients from the skin during cooking, but it's not bad that it cuts down on the work too. I just toss the quarters in a pot with a few cups of water and simmer, stirring occasionally, until totally squishy (adding more water if necessary). When it's all soft we put it through the food mill. I found this at Goodwill for $5, which was quite a steal, but I sure wish it was one of the big ones. Pre had a great time trying, but it was tricky for her to know when to reverse for a while to pull up the peel and such.
We ate a pint immediately, and canned the rest. This stuff is sooooo good. I can't believe people add sugar (or Heaven forbid high fructose corn syrup) to applesauce!