I know that every directory starts with / because everything in the system is below root. That's why I type /before I go to my pictures or desktop or what have you. Trash has been bugging me though, it doesn't start out with /. It is trash:/// Why is that?

Because thats how kde accesses stuff in the file system. trash is actually ~/.local/share/Trash

If you use Konqueror click View > Show Hidden Files, which enables you to see directories that begin with ./

That's a URL for KIO, not a directory path. KIO is KDE's infrastructure for providing a uniform interface to resources of various types. Basically, there's a KIO slave that knows how to read and write things using each of the different protocols, so an individual KDE application just has to do its file access through KIO and it will automatically support all the protocols KIO understands, which includes local files, SSH, FTP, SMB, etc.

The trash:/ protocol is one of several KIO slaves that just provides a specialized view of the local file system. There are several others, such as applications:/, system:/, and media:/. The actual location of the trashed files is defined by the FreeDestkop.org trash specification, which, as AceofSpades19 said, puts the files in ~/.local/share/Trash by default.