Make sure to tell how you would keep the pet alive as in if it needed special living conditions.
Personally I would have a penguin. I would keep my basement nice and cool and it would live down there. During the winter I would let the penguin go outside. I would have to feed it lots of fish though.
I would want a domesticated large terrorbird, on which I could ride around. Titanis was about 2.5m high, that seems enough. I would look so much better than those riding on horses I would keep it in a large enclosure and feed it with raw meat. I don't think I could find live food for such a big animal...