Asteroids are discovered at a rate of about 2 per day. Here are some pointers:
Asterank has compiled hundreds of thousands of sky survey images taken by publicly funded observatories.
To discover an asteroid, watch the animation of the night sky and look for a moving white dot.
There's a good chance that moving dot is an asteroid. The first user to notice the dot gets potential discovery credit and naming rights.
Genuine asteroids are light gray objects that move around when the rest of the sky isn't moving. The asteroid is in the center of this image: