Lol. When you have a cousin whose name is Vera: «Eu gosto da prima Vera.» XD
do = de + o ("of the" for masculine words) da = de + a ("of the" for feminine words) They both are used for singular words.
da primavera==the spring so why why do you miss out the "THE" in the translation
I think that it's similar to French. With a generality and a verb of appreciation "Eu gosto da primavera" could mean either "I like spring" or "I like the spring", but a native speaker would have to confirm.
Sometimes in Portuguese when we say do or da it lit translates to "of the" but the of is omitted most of the times and sometimes the is also omitted.