I rubbed my hand against the faucet and it was dirty cleaning grime off of the sink. Do I need to clean it, or is it true that the running water will flush any bacteria? And how do you clean the area where the water comes out of the sink if I do need to?
I appreciate any feedback. I thought that a plumbing forum would be appropriate, and I would receive the most honest answers. Thank you.

I assume when you say "water out of the sink", you really mean water out of the spout on the faucet. You can unscrew the screen. Grab it with your hands or channel locks and screw it counter clockwise. You can disassemble it and clean it out. Be sure to block the drain with something in case you drop a part.

As for general cleaning, use a bathroom cleaner to clean the faucet and sink. Something that won't scratch the finish.

On the end of the faucet there is a removable attachment that keeps the water from splashing in the sink and filters the water. That's the aerator. Turn it clockwise (as you're looking down on it) to unscrew it. clean the faucet properly and filter is also clogged with particles, clean it also. Carefully tap the aerator on the sink and wash out the particles. Be careful not to lose the parts (washer, screen, and water restrictor.) Then, screw it back on.