Will a cherry tree grow from the pit?
If so, will be it the same type of tree as the fruit came from?
You will get another sweet (or sour) cherry but it may have different fruit from the parent tree.
I can't answer about sour cherries, never noted them to grow from any of my pits. I have had sweet cherry seedlings grow, but they revert back to the wild species, Prunus avium, bird cherry. Fruit is much smaller, thin-fleshed, same size pit, tends to have a bitter flavor. Makes a nice ornamental, fast growing, and you can make a nice jelly from the juice, but it's not great in terms of eating quality.
I know they must grow sweet cherries somehow from seed in breeding programs, but any I've grown from seed have been the bird cherry type.
It will, but chances are it won't be any good. There is just one (I think) sweet cherry that's self-compatible, so pollinizer trees are planted in orchards. These are often not suitable for eating, and that's why the fruit you get if you try to grow out a tree from fruit you bought in a store is crap.
so, dennimi, the seedlings don't really revert to the wild species. the reason the cherries that result from crosses in breeding program are (with luck!) good is because the breeders are making focused, planned crosses between two genotypes with desirable traits.
Tart cherries can self-pollinate, so chances of getting something good are better from it, but it won't be the same as the parent.
The odds depend on the specific genetics involved in each instance. It is often stated, as here that seedlings of cultivars automatically have inferior characteristics. That is not necessarily the case. Many new cultivars of various garden plants are based on seedlings that arose. Many of these were not the result of intentional crosses.
I have eaten many cherries from seedling trees [Bird droppings], in Switzerland, along ditches and forest, a little small but good!