Giving two week notice and taking PTO?

I was hoping to get some clarification as I look into this further. This is my first full-time job so I'm not sure on where this stands.

I'm a full-time employee and have received a job offer somewhere else. Unfortunately at the end of this month I have 6 days vacation booked, and my new start date is about 12 days after my vacation ends. I don't want to push my new start date, can I give a two weeks' notice while on PTO? If they choose to terminate me immediately would I be paid out for my remaining vacation days or any severance? TIA