I put my notice at my job and I'm going on vacation during that time of notice. Do I get paid for vacation and the remainder of the days that are left?

You put in notice and set your final date to after the vacation is used up.

I gave notice and my current employment ends the 17th. I start my new job the 20th. I'm on vacation from the 15th -19th

Assumning you work Mon-Fri that makes sense.

It depends on your vacation policy. At my last job we would pro-rate your vacation when you left - so lets say you left in March and before you left you used all of your vacation time well you would actually OWE the company days. Be certain to check your employee manual very carefully. Because you may end up with an empty last check.

Glenda's got it. Its different from place to place, so check the manual or call HR/Payroll to double check.