I have done some research and know that you can breed pigs with carrots. How do you get carrot seeds to plant carrots?
The carrots themselves are the seeds. You can just plant the carrots you have, then they will grow and "duplicate".
Unlike Wheat, planting carrots requires that you have carrots already, which is kind of a catch-22 if you have none to start with.
Carrots were added to PE in 0.8. (December 2013) as a rare drop from Zombies.
Once you have a single carrot, planting it and then harvesting it when fully grown will drop 1-4 carrots, so from that point onward you're likely to get them faster from farming.