I believe I understand the points about "Ki tudja, hogyan ..." and other examples, but I think the matter of Duolingo's translation is still open. When I mouse over "tud," it gives "knows" as a translation, with the word "ismeri" (meaning "to know," as well) before it in parentheses. Is "ismeri" being presented as a synonym, but only in specific cases?
A naked "tud" usually implies ability to do something. When talking about "knowing how to", it would be "ki tudja, hogy(-an) kell festeni?" Finally, the verb "ismerni" means "to be familiar with somebody/something". I think its safe ro say that if the verb "know" refers to a noun, and not an action, then you use "ismerni".
This is true in most cases but not a rule by any means. "I know math." If you translate it as "Ismerem a matekot", it sounds like "yeah I've met it, I know some of its things, I could describe it in a few sentences". "Tudom a matekot" means you have a considerable amount of knowledge at math. So one would probably aim for the second.