After living with college roommates that couldn't seem to do the dishes properly, I realized I had to get the word out there about proper dishwashing duty.
Step 1: Dishsoap and Sponges
The first thing to realize is that dishwashing is not about sanitizing (killing the bacteria and viruses on) your dishes. If you look on your dishsoap (any brand will do), you will notice that it does not claim to "disinfect" your dishes!
What is the point of dishwashing?
To remove food, scum, scents, flavors, grease, oil, etc. from your dishes.
What is the best tool to use?
Use a sponge that has a scrubby side on it. This way you can soak up soapy water to scrub away at your dishes with. Using a plain sponge with no scrub is pointless - it won't scrub the food or grease off of your dishes!
How do I get rid of the "germs?"
Washing your dishes will help get rid of most of them, and the rest tend to die once the dishes dry off (bacteria can't live without water!). However, if you're concerned, you could always rinse your dishes with a 1/20 bleach dilution, or add a tiny bit of bleach to your dishsoap.