To make someone in your town mad at you, walk into them for a while and push them around. If done correctly, they will become mad and steam will shoot out of the top of their head, but won't remain mad for too long.
To make a neighbor mad, hit them repeatedly with a net. This will take some time, but if done correctly, steam will eventually shoot out of their head and they will yell at you.
When an animal has just moved in, keep talking to them. They will eventually say things like "Can you wait? I'm still unpacking!"
Keep talking to an animal that really wants to move and they'll eventually say something like "No! I've already made my mind."
Push animals around a lot, especially into ponds, rivers, or the ocean.
When animals ask you to deliver an item, they will say, "Better not open it!". Open it to make them mad at you.