I play Civ4 with my girlfriend. We always play multiplayer games, just us two and 5 AIs, and we play nice (ie. we only fight with the AI, but we aren't allies). I play on deity and she plays on prince. (because we are on different difficulty levels, the AI isn't a deity AI, so it isn't that bad.)
She has no interest in single player, and she doesn't usually play to win – she's happy enough to just build up a decent civilization without aiming at a particular victory condition.
I personally like to learn games by playing on my own, but I know some people don't really have the motivation or interest to learn complex games in that way. So I suggest that you ask your girlfriend how she would prefer to learn the game, and if she doesn't really have a preference, then I suggest you play like I do – friendly multiplayer.
Also, I suggest that you try not to overwhelm her by talking about advanced strategies and techniques. Just tell her the very basics, and tell her that you'll give advice if she wants it, and then let her just play her own way.
[edit]
If you do play multiplayer I suggest that you don't be permanent allies at the start. Allies share research and diplomacy, and I think she should be in charge of all of that stuff while she learns. So I think it's better if you play as independent civs. If you need to, you can gift her military units and tech if she looks like she's really in trouble.