You actually have several ways you can move around -- after you hit the arrow button beside the filename textarea. Do this first.
Double click on a folder and it will open.
Highlight a folder and type Command-down-arrow to open it.
Type Command-up-arrow to move up a directory level.
Switch to Columns view (click the button or type Command-3).
In the Spotlight search field beside the current directory name, type a folder name. I find this one flaky but sometimes it will present you with the folder you wished to find. Double click (or use any of the above methods) to open it. Note that if you highlight an item, at the bottom of the results list you can see the path to that item. You can double-click on any of the folders to go there.
If you have a Finder window open and visible, you can drag something (a folder or a file within it) and then that directory will become the save location. If you click and hold on the window icon (in the top bar beside the folder name, you can drag that as well. You have to hold the click until it goes dark. If a document is open nearby, you can even drag its icon from its document bar. The directory that the document is saved in will become the save location.
Command-Shift-G offers a 'Go to the folder' option. You can type any path (you even can use auto-completion) and then clicking Go will make that the save directory. This works even if you don't change the Save dialogue's mode.