I have a few suggestions here on helping your girlfriend to like riding:
When I met my last bf that rode, I was just starting to get into riding. I had a road bike and was averaging about 11mph, thought 20 miles was infinity and hills were evil.
4 years later, I suck way less, ride centuries, adore two mile hills and am thinking about racing this year. I also ditched the bf at some point
The guy I was dating talked me into going to Ragbrai that first year. So he took me out and rode me up and down the river valley hills. But we went at my pace and we rested when I needed to. We went on several organized rides in Minnesota to get ready as well.
I highly recommend organized rides as a way to expose your girlfriend to other cyclists. This will spur any competitive streak she has even if she denies that she has one. I can tell you that I never walked a single hill after I started riding with groups. I'd rather fall over dead from a heart attack than admit that I can't ride up a hill. And there's at least one state park hill in Iowa where it was touch and go for a minute
It's a lot of fun to meet other people who ride and to be exposed to the sheer diversity of the cycling demographic.
I also recommend getting her off the paved trails with the kiddies and family riders. Whether you're mtn or road riding, get out to some interesting scenery, rolling country roads, diverse trails, whatever.
As for relationship advice, do what everyone else says. Encourage her, offer to pull a lot, remember that, while some chicks get faster than guys, we just have less power in our legs than you as a general rule. Don't treat her like a baby. Some of the condescending crap I've read on this thread makes me wonder how some of these guys even get girls to talk to them!
If she likes it, she'll continue to do it at some level. It takes a special kind of wacked chick to love her bicycle as much as some of us do though. Keep in mind that a lot girls never get that far...