What is the best way to move animals long distances?

Boats

You can also shove them through a nether portal

How do you get an animal into a boat? Just push it into one?

If you place a boat on a block it is standing on, the animal will pop right into it

Oh cool. Good to know, thanks!

@WBlaine If you place a boat on a block it is standing on, the animal will pop right into it

congrats on mod m9

You can also row boats on land as long as the path is level (completely level) or down hill. Great way to get villagers too.

Minecarts and Merged Minecarts are also options depending on your needs.