If you are careful, you can also save Advanced Settings. Find the GameInfo folder, typically here:
C:\Program Files (x86)\Steam\SteamApps\common\sid meier's civilization v\Assets\Gameplay\XML\GameInfo
There are two xml files of interest:
CIV5PlayerOptions.xml
CIV5GameOptions.xml
Be sure to save a copy of these files before modifying them. The CIV5PlayerOptions file already has default commands, which you can just modify. On the CIV5GameOptions file, you will need to add the default command. As an example, in CIV5GameOptions, you will find the Quick Combat option.
<Type>GAMEOPTION_QUICK_COMBAT</Type>
<Description>TXT_KEY_GAME_OPTION_QUICK_COMBAT</Description>
<Help>TXT_KEY_GAME_OPTION_QUICK_COMBAT_HELP</Help>
Then add the indicated line to set the default.
<Type>GAMEOPTION_QUICK_COMBAT</Type>
<Description>TXT_KEY_GAME_OPTION_QUICK_COMBAT</Description>
<Help>TXT_KEY_GAME_OPTION_QUICK_COMBAT_HELP</Help>
<Default>true</Default>
Of course, you'll have to restart the game for the changes to occur, but then you should see the option already checked in the Advanced Settings Menu.
(That setting should be part of the global user interface, and not set per game.)