So I typed доброго вечір, and now I see that version of "good" doesn't go with вечір. What's the difference between all the versions of good? Is it the case thing? And Duo tells me доброго вечора would've been fine. What's the difference between those two words for evening, or is the same as with all the words for good?
We have two forms of "good evening": добрий вечір and доброго вечора. Both are common. In a first phrase both words are in nominative. Second uses genitive because it's like a short for "I wish you a good evening". Adjective should agree in case with a noun to which it relates. You used one word in genitive and one in nominative which is not right
As far as I know, adjevtives change depending on the gender.
добрий вечір (good evening)
добра дружина (good wife)
добре місто (good city)