З ким ти хочеш мантрувати? - In my opinion that's correct too.
1) мандрувати is accepted in this exercise.
2) want ≠ like, so хочеш is not accepted
"мандрувати", actually, but I do agree with you. It's the first word that comes to my mind when I see "travel".
з ким любиш подорожувати? - should be correct..
In standard Ukrainian we usually don't skip pronouns. In spoken Ukrainian in some cases (depending on the sentence, the speaker, the region, personal style etc.) one might skip the pronoun.