Hi Nat09301,
So many people beginning a diet and/or exercise program are under the false impression that strength training will make them bulky or that they would benefit from "losing weight" first, then, when they slim down, begin the strength training. The thing you need to realize is that strength training is actually a critical component to losing weight, and promoting lean muscle mass. The more lean muscle we have, the more calories and fat your body will burn around the clock - even at rest. Most bodies will not gain the kind of muscle that will make them appear bigger (it's actually quite difficult for most people to put on quality muscle). The kind of strength training you are going to be doing is not designed for that, and will likely make your body appear smaller and more compact, especially as you continue to burn fat.
And, the most effective way to increase muscle mass and burn fat is through a combination of cardio and strength training exercises, and you are doing great with the cardio. I urge you to start participating in the strength training classes your gym offers now. Try to get in two total body sessions per week - spaced out by four days. On your other days, keep up the cardio.
Keep up the great work and keep me posted.
Good Luck,
Paul