I'm looking for a way to create new albums within the Photos app on my iPhone 3GS, preferably without having to connect the device to iTunes. Is this possible?
No, it's not possible to create new photo albums in the Photos app on the iPhone. (Not on my iPhone 3GS running iOS 4.0.1 at least. And it's not possible on my iPad either.) See http://discussions.apple.com/thread.jspa?threadID=1413776 and http://www.mac-forums.com/forums/ios-apps/200714-creating-photo-albums-ipad.html.
This Apple Support FAQ describes how to sync photos: http://support.apple.com/kb/HT1389.
I wouldn't be surprised if Apple adds this feature soon. I expect they're working toward making the iPad work completely independently (without syncing), and the iOS (and app) features they add will likely also be added to the iPhone.
iOS 5 and later supports creating new albums. However, the albums do not work like folders in a file system, but rather like tags or groups, because when a photo is added to an album it still physically exists in the camera roll and is linked to from the new album.