Exactly--it's an idiom. That makes it difficult to get the translation right the first time, but now you know. (Digression: One of the things I don't like about Duolingo and other conversational/immersive language curricula is that new words, grammar, usage, idioms, etc. often aren't explained before they are thrown at you.)
Not a native speaker but Linguee.com has several translations for "andar por aí": go out, go around, go round, go about. The examples seem to be more like saskia.schroeter's first use, "I have been walking around [aimlessly]", rather than "I have been walking around [the tree]". See, http://www.linguee.com/english-portuguese/search?source=auto&query=andar+por+a%ED.
Have been walking around is the ultimately best answer.
The other would be ok when adding a duration to the present perfect, with the sentence still true today:
- I have walked around for years = Eu tenho andado por aí faz anos
- I have walked around since I left = Eu tenho andado por aí desde que parti