I'm about 20 lbs overweight and have taken up tennis, lap swimming, and Zumba as a means to shed the extra pounds. My biggest concern about losing weight is my breast size. I currently am an A-cup, and I worry that I'll drop down to a AA if I achieve my weight loss goal.
Can someone give me an idea of what to expect? My mother always told me you lose weight in the chest first, but I'm not sure how accurate that is.

Breasts are made up of breast tissue and fat. Some people have a higher percentage of tissue and some have a higher percentage of fat. If your breasts are mostly fat they will shrink when you lose. If they are mostly tissue, they probably will not change that much.

As far as YOUR particular breasts, I can't really answer that question Only time will tell.

You will probably lose some boobage, sorry to have to tell you But as Rebound said, it's different for everyone... I have noticed a few inches in the chest lost since January, but I'm still wearing the same bras, and I lost a lot more in the waist than the boobs. Inch loss can just as easily come from back fat as boob fat. And not everyone loses inches in the same way/distribution. There's really no way to know for sure!
Pushups/chest press may help compensate with pec muscles, I do a lot of those.