No, no, Throbert McGee. I just imagined myself in this situation in the driver's place and these two phrases would seemed a bit bizarrely to me. In my view, the words "Отвезите" and "Довезите" are superfluous here. These phrases with them could say a child of 5-10 but not an adult. I think so... It's the same like this:
A: What is your name?
B: My name is Brad.
A: What is you surname?
B: My surname is Pitt.
A: How old are you?
B: I'm 49 years old.
A: Where are you from?
B: I'm from the movie Fight Club.
More naturally the answers would sound like this:
A: What is your name?
B: Brad.
A: What is you surname?
B: Pitt.
A: How old are you?
B: forty-nine (years old).
A: Where are you from?
B: from the movie Fight Club.
Something like this, you know.
I'd say your second option, but! I'd add "вчера", without fail. So the sentences would be like this: "Я вчера ездил на такси на Красную площадь". Your sentence sounds correctly but, with "вчера", it sounds more euphonious.
And, in conclusion. It's what my behaviour would be in the situation like this...
I flag down a taxi, open the door, stoop and say to the driver:
Добрый день. До Красной площади довезете? ~ Hello! Can you take me to Red square?
Он: Да. / Угу. ~ Yes, I can. / Uh-huh.
Я: А за сколько? ~ How much?
Он: (За) 700. ~ 13.5 pounds
Я: OK! ~ OK!