After a lot of testing, I managed to get a Civ into the game with full functionality.
Certain things tend to happen depending on what you do wrong. The first thing to check is to make sure you've added the actions to import all your XML files. The art files (in my experience) don't need to be imported (with "OnModActivated" and so on), but you must make sure they're listed in your mod (they should be listed on the right next to your XML files in Mod Buddy). They, along with your XML files, must also have that "Import into VFS" setting as true (as others said).
If that doesn't work (and your modded stuff doesn't show up in the game at all), you probably made an XML error. One critical error can make it so the whole mod doesn't show up.
It's always good to make one big change at a time, test it, and then move on. If you make a bunch of changes and one is broken, you have to check all of them to make sure you've fixed the problem.
If you're making a Civ, copy another Civ and edit it. Leave the references to the existing Civ in until you change that part. (Also, if the Civ or Leader you copy had the <ID>0</ID> tag, I'd remove it unless you delete all the other Civs. That tag tells the game to start counting the ID numbers of the Civs/Leaders and leaving that tag might stop your mod from working or it might overwrite something already in the game. I'm not absolutely positive, but removing that tag didn't cause me any problems.)
The hardest part of modding Civ V is just getting things to work.