Can I get smaller tires without changing my rims? I currently ride a hybrid that came with 700x32c tires. Can I swap out a narrower tire without changing the rims, or do rims only fit a specific size of tire?   How small can I go before I have to buy new rims?  I hear having narrower tires makes you go quite a bit faster.  Does it really help that much?
Most 700c rims will work with essentially any size of tire. Yes, smaller tires generally reduce rolling resistance a little, but unless you're road racing, it's rarely worth going below about 700x23c, IMO. Also note that just being smaller doesn't necessarily mean lower rolling resistance -- one brand of 700x19c might have higher rolling resistance than another brand of 700x20c, for example.

The big change (if any) happens when you switch to slick tires (and most 700x32c tires have fairly substantial tread if not outright knobs).