In my classes the syllabus is the contract for the class, and needs to contain statements about cheating and the consequences. My dean had me add a line, which I originally set as “the student receives a score of zero for that item.” Since then, I’ve added “… and the final grade is reduced by two letter grades” to further drive the point home.
I give my students paper copies of the syllabus, which we go over in detail (since it’s often many student’s first college class), and a first-day quiz which includes “I received a copy of the syllabus and understand its contents and agree to its terms.” They cannot continue taking the class without getting that question right.
At my college I’ve had a couple cheating cases, and our Dean of Academics and the Dean of Students get notified of them. In both cases they offered their full support.
I’ve gone to proctoring exams by walking around the room, standing in the back and just watching (actually I’m taking a short standing nap). I also have had two versions of the same exam with the questions slightly mixed up, but that was too much work for me. I don’t let them keep their exams (since I reuse them for future semesters), which is our usual practice.
My bottom line: don’t sweat it this time, but keep an eye on those students in the future and when proctoring. It’s part of the learning experience, especially for you. There will be other challenges, I’m sure. Welcome to college!