My BMI is 19.5 and I feel quite thin so I am trying to gain weight. What foods should I overeat without affecting my health? I.E milk? Protein powder?

Why? Why do you want to increase your weight. A bmi of 19.5 is healthy. Once you start consuming more, even healthy foods, and gain weight, you may end up weighing more than you may have bargained for. Eat a well balanced diet rich in fresh vegetables, fruits and low in meats.