I would look into Flickr, Smugmug, and Zenfolio. I'm sure there are others that are good, as well.
Flickr is pretty much everywhere. Many, many photographers use Flickr, and many people know about Flickr. So you'll get a lot of exposure by posting your photos on Flickr. You can get a "pro" account for just $25 a year.
I, personally, use Smugmug. It seems weird, but I feel that my photos just "look" better. Part of it might be the black background. I'm a sucker for photos on a black background. They just look more, I dunno, classy and professional. I also like the way that my photos can be organized in Smugmug. You create different galleries for different photos...probably similar to what you're already doing on a computer, using folders to organize your files.
I see a lot of photographers using Smugmug, too. Their screens are highly customizable, too, so you can make it look like your own personal Web site / portfolio, without a bunch of Smugmug logos around it.
With Flickr, I'm still getting used to having a one, long, continuous "Photostream". I know that you can organize the photos into different galleries, too, but the Photostream kinda throws me off. If I used it more often, I'd probably be more used to it.
Zenfolio is a close competitor to Smugmug, and I believe their prices are a little lower than Smugmug's. I haven't looked into Zenfolio in a long time, so I'm least familiar with them.