What's the location and name of the save game file for Giants Citizen Kabuto?

They are found in the game's install directory, C:\Program Files\GOG.com\Giants – Citizen Kabuto\savegame by default. Within that directory is a profile directory and a SaveGame# directory which contains the actual save files, keymapping.sav (your keymapping settings) and mission.sav (the game progress save).

In Program Files I have a folder called Giants which has folders called Bin, EReg, Music, Stream, and Stream English. There are also a few dll files, the readme_english file,the Giants.exe, the Uninstaller, and Giants.mpi. I have V1.1--is it different with different versions? I still can't find mission.sav.
Another web page says "The save file has the extension gzp"--for Gazelle system page.
http://www.file-extensions.org/filetype/extension/name/game-files
I found some gzp files, but I don't know which one is a save file.

The GZP files are not the save files, those are the game's data files (graphics. missions, etc.). The save file directory is not created until you actually run the game and reach a mission checkpoint, so if you don't have it, you likely haven't played the game yet. Game versions don't change where it saves the games.
Since you are using version 1.1, I'm assuming you don't have the GOG version (it's version 1.4). In that case, even though it won't do anything for your save file issue, I recommend you at least update with the latest patch.
EDIT - You aren't possibly running the game on Vista, are you? If so, it might be putting the save file directory under the "Compatibility Files" option.

I'm on Vista and most of the way through the Meccs section and can load the game
anywhere up to there, so the games are being saved somewhere. An Internet
article says to press a button in Explorer to see Compatibility files, but I
don't have Explorer--I have Firefox, and I don't see a button for Compatibility
files(?).

It's not in Internet Explorer, it's in Windows Explorer, the file browser you use to see the files on your computer (i.e. "My Computer").

I found it!!! The Microsoft article on "File and folder Virtualization" said
to go to the C:\Users\User_name\AppData\Local\VirtualStore\MyApp folder. But I
didn't have AppData listed there by clicking my way through the usual way (?).
So I put AppData in Start > Start Search and found it.
http://support.microsoft.com/kb/927387/en-us
Then I used the rest of the article directions to go to Virtual Store, but I
didn't see a MyApp folder (maybe I'd find it if I used Start Search)--I found
two other folders: Program Files and Windows.
The Program Files folder had folders of nine things, including four games I'd
taken off my PC a while ago and Giants. In Giants was a folder called savegame,
and in it was a folder called (User name) and in it were keymapping.sav and
mission.sav.

Yep, you just found the long route to the "Compatibility Files" option. The reason you couldn't see the AppData directory is because it is a hidden directory by default. In order to see it by simply browsing, you would need to change Windows Explorer options to show hidden files and folders. The "MyApp" I believe is just shorthand that Microsoft uses in place of "insert name of application/directory here".

In my experience, it isn't sufficient to back up the savegame directory. I have tried moving Giants profiles from one machine to another, and that doesn't do it.

There is also a registry entry (in C:\Windows\regedit.exe) under ... I forget right now but I'll check later. Something along the lines of HKEY_CURRENT_USER\PlanetMoon\Giants\Profiles\<YourName>. You need to back up that whole registry folder and move it to another machine or back it up, to restore your saves.

Is there an easier way?

I recently had to get my old (well not so old, but anyway) savegame back, and I too discovered it was not sufficient to simply copy the savegame folder. For unrelated reasons XP 32-bit will not boot atm, so I had no way of accessing the windows registry either. I also have Vista 64-bit (dual booting, well, triple booting with Ubuntu), so I installed the game in Vista.

What I ended up doing:

- Created another player with the same name as my old player name.
I then played the first mission, but I do not think it is neccesary (might be to see the location of mission.sav though).

- Navigated to the new savegame folder.
I also noticed the "mission.sav" file was now in a subfolder of a different name than in my old installation of the game.

- Replaced this file (mission.sav) with the one from my old savegame

And that worked :)

Thanks, this method worked for me.

It is not necessary to complete the mission. Just starting a new game would create the "mission.sav" file in windows that can be replaced with the old savegame.

I also noticed that once GOG saved the "mission.sav" savegame file in "GOG Galaxy\player\savegame" (player was the default player name in the game). Only after a reinstall did the game actually create a "savegame" folder in the game folder. So be careful.

It's been years, but If it can help someone that may want to replay the game again, on my gog (not Galaxy) version of Kabuto (the latest offline installer) , i installed the unofficial patch 1.5
After setting up the game, my keys and various options i played for a while.

As i usually back up my savedgame (so if i uninstall and reinstall i don't have to redo my keybinding and etc...) i was looking for the saved location, and didn't found it anywhere people said it was.
I guess things have changed a lot since those 2009 replies (unless it's the patch unofficial 1.5 that changed it).

I found that my saved game/keybinding are now found in :
C:\Users\MyUSERName\AppData\Local\Giants\

"MyUSERName" being your user name of course, so just back up that Giants folder once you have created your keybinding and setup your ingame options.
Remember also to back up the registry entry (access it with regedit)
HKEY_CURRENT_USER\PlanetMoon\
As previously mentionned.

Windows has changed.