You don't want Save or Export... you want Render. (although I realize that you probably want to save your Project too)
If you are in one of the right places where the menu item appears you'll see Render across the top of the screen.
Under that the dropdown will have Prevew Settings, Preview, Render Settings and Render.
Open Render Settings and change the options to the desired format (AVI or MOV) or as sequential images (PNG, JPG, etc.)
Choose a filename too.
Then go back to the Render menu and this time select Render.
This will create the frames in the location you specified in Render Settings.
I'm sure there are (or can be) some Shortcut keys for all of this.
To convert to a format such as GIF you'll have to use another program. OpenToonz does not support that format.