Dictionaries are all well and good, but all too often a term will not have the same nuance in a different language even if the dictionary definition is similar. The best way to learn what a word really means is to see how it's used in context.
"Nag" does mean "harass", but in a kind of familiar way, as one family member to another. You would not nag a lion -- you can only nag a person, and even then, usually only someone fairly close to you, who you see regularly over a long period of time. A mother can nag a child, or a child can nag a parent. Sometimes a friend can nag another friend.
But any other relationship and we would probably use a different word.
Of the remaining choices, while it would be inadvisable to bite the lions, tease makes more sense in context. Fret doesn't fit at all. As you say, annoy would also work, or harass, or bother, but these aren't choices.
As for "to", it's simply the correct preposition to use for these kind of activities. We go to the zoo, or to the park, in the same way we go to a store or to the library.
Yes, it does make perfectly logical sense that the sign should make the request not to the visitors coming to the zoo, but rather the visitors who are already in the zoo. Unfortunately, that's just not how it's usually phrased.
Some other examples:
Visitors to the National Museum are asked not to touch the exhibits.
Visitors to Hollywood should know that they are unlikely to see a movie star just walking around.
Visitors to the gallery must have an appointment.