I`m 19 years old and weigh 128 pounds and I`m 5`9``, Which makes my BMI approx 18.9, which is healthy, I know. Lately I`ve started going to the gym a lot and eating better in hopes of losing weight around some specific areas on my body, hoping to get down to about 120 pounds. This would make my BMI around 17.7, which apparently is underweight. Is this a healthy goal weight for me?

You don't need to lose weight. A BMI of 18.9 is on the low side of healthy, and 17.7 is definitely low.

By all means go to the gym, but work on building muscle, because at your weight, you risk some of the problems of the chronically underweight unless you make a special effort to build muscle mass and bone density.