Eva,

Coming from a former gym rat, I would say weights are really good for you. If you want to look better and be able to lift your bike atop the car easier, upper-body lifting is important. But just for cycling, I would say leg and core strength are way more so.

Do you do yoga? A simple plank pose will work wonders for core strength. Lifting your body weight can be plenty if you do lunges. "Link" those lunges: Step from one lunge, balance on your front foot and step down to the other foot, and your lunges magically become a balance exercise.

BOSU (half a balance ball on a platform) is AWESOME for balance. Just try standing on it at first (either side), then squats or moving your arms around.

Being pretty young, I thought my balance was decent, but when I began skate skiing two years ago, when you MUST transfer all your weight to one leg and glide for a couple seconds before easing back to the other ski, my mountain biking improved drastically.

It's rare that a sport is totally independent. IMHO, if you cross-train, you'll be more athletic, and likely be better at all your sports. And don't worry about "bulking up," that's much more to do with genetics and big weights than normal weight training.

And it's totally natural to swerve on the bike when you try to turn your head or move your hand. You just have to learn what happens and then plan for it and eventually you'll be all over that bike without it changing course.

Good luck!