Is this normal? Is there a way to have the game use all (or at least more) of my screen?
Thanks to anyone who can help!

Runs full screen for me. It sounds as if your PC isn't switching into the correct res for the game, and is just staying in your native desktop res. Apart from the cliché response of "try updating your drivers"...
I guess you could try lowering your desktop res, though IMO that would be more of a workaround than a proper fix.

I haven't been able to dedicate much time to Freespace yet, but I fear the game only runs at a resolution of 640x480.
If you have an LCD display with a native resolution higher than 640x480 (which is quite likely nowadays), the game may appear in a tiny box at the center of the screen if either the display or the graphic drivers are set not to scale/stretch the image in lower-than-native resolutions. You should be able to change this and having the image scaled to fill all or at last part of your screen, even if this will most likely result in a slightly blurred image (with higher resolutions avoiding scaling is generally preferable, as it doesn't reduce the image quality).
Ideally, I'd first check your graphic drivers settings: you should look for an option to either use your display built-in scaler or letting the driver do the scaling, preferably with fixed-aspect ratio (resulting in black bands on the side of the screen but without stretching the image if you're using a widescreen display). Which option is better depends on the quality of your monitor scaler and therefore it's something you'll need to test by yourself (by default nVidia drivers should be set to let the monitor do the scaling I think, and I'd start with that option).
If you have trouble finding these settings, please let me know which graphic card you're using.
If you choose to let the monitor do the scaling, check your screen OSD menu using your screen controls to see if you find an option that zoom/scale/stretch the image when using non-native resolutions: again, ideally you should choose the scaling option, if available, that let you keep the same image aspect ratio - 4:3 in this case - to reduce stretching.
Unfortunately it's not rare to find the various scaling modes called in different ways in different screens/brands OSDs, so I can't be more specific, but your monitor manual should hopefully help.
Hope it helps,
!K

Wow!
Thanks a lot for the help!
I'm at work right now so I can't try it out but hopefully I can play some Freespace when I get home tonight (and not in a tiny box).
That's a lot of great info, Kismet.
And, Fluke, it sounds like if all else fails your solution will work.
Again, thank you both for the great help.
EDIT: And it was just one option on my monitors menu. I feel dumb. Thanks again!

Glad to hear you've been able to play it in a bigger box :)