Should i go to primary care physician or to university counseling center for mental health care?

Yes: Seek help from either place. Both can provide assistance.

Counseling center: Will save some time, although you could consult with your physician if you want to go privately and want a referral. If there are medical issues as well as emotional ones, that might be a reason to consult with your doctor first. Otherwise, the counseling center will get you started in therapy sooner. Good luck and let us know how you do.