Am I thinking about taxes the right way

Just did my taxes for the first time, I got $2 refund. Which is what I read is the closer to zero the better, is that right? My dad said I should contact my hr/payroll to have out more money out of my income so I can get more back but that doesn’t make sense to me since I put a good amount to my savings each paycheck and calculate my bills accordingly.

As I move up in my job and get raises will I have to adjust my withholding personally or will it automatically be done? Sorry for the ignorant questions.