I'd say walk the walls - you get to nose in on some of the big old houses in around (and presumably accommodating clergy etc.) the minster as well as get a good view of different parts of the city.
Given that you're 16 - Evil Eye Lounge is a great place to chill out with a coffee or non-alcoholic cocktail, especially if the weather turns bad
It's on Stonegate, also the home of little Betty's - go hear if you fancy parting with a tenner for tea and cake.
I concur with marinyork on the Sam Smith's pubs, especially the one down by the river. That reminds me, funny story.
The Sam Smiths pub near the entrance to the Minster on petergate, is basically a gay pub. It's not overly obvious because it ain't exactly Manchester or Birmingham is it! They had the best ever food blackboard outside I have ever seen in my life.
"This week is National Sausage Week. Go on.... be daring, put a sausage in your mouth"