I'm not sure, but I doubt I've ever said, "I rode a taxi," or some version of that. Maybe something like, "When I was riding home in a taxi last nite, I saw an accident at ~." Or, "When I was riding home in a taxi last nite, I got sick and ~ ."
I would also be more likely to use take/took as opposed to ride/rode but I would use ride/rode as well. As for your question about rode I would say normally if you are the driver you would say drove as opposed to rode but if you are a passenger you would be more likely to use rode. "How did you get to work this morning?" "I rode with Harry he lives near me so it wasn't out of his way."