amount of calories

You'll likely lose weight but you won't gain muscle with that deficit. In fact, you're likely losing a high percentage muscle with your current methods especially if 1200 isn't your net calories. In nearly all cases, you only gain muscle during a calorie surplus. You might get stronger but you won't gain muscle. It's why people go through cut/bulk cycles to put on more muscle.

I really like this thread because it talks you through the two main ways to set your calories (MFP & the TDEE method): http://www.myfitnesspal.com/topics/show/819055-setting-your-calorie-and-macro-targets

So I am currently 22 and trying to get back into shape. I have been working out now for 6 weeks and have already lost 13 pounds. I am 5'5" and weigh currently 157. I have been trying to stay around 1,200 calories a day. I burn about 500-600 calories each day with my workout. Is 1,200 calories the right amount? I am hoping to lose about 15 more pounds and gain more muscle.

MFP gave you the lowest default value.....1200

This is not a reflection of the number of calories your body needs (your height, weight, age, etc).....but rather a reflection of "I want to lose XX pounds a week"

Keep in mind 1200 is a net number....that means if you exercise for 500....you eat 1200+500. This is because MFP gave you a deficit with ZERO exercise. That way people who can't/won't exercise ...still lose weight.

While eating at a deficit your goal should be to MAINTAIN muscle. You can maintain quite a bit of muscle....provided
1. Your deficit isn't overly large (1200 is likely a large deficit)
2. You strength train
3. You eat enough protein

Make sure your weekly weight loss goals are moderate

Pounds per week
75+ lbs to lose 2 lb range
Between 40 - 75 lbs to lose 1.5 lb range
Between 25-40 lbs to lose 1 lb range
Between 15-25 lbs to lose 1 -.50 lb range
Less than 15 lbs to lose 0.5 lbs range