In vasovagal syncope what does your heart rate do, go up or down or no change?

Varies. Usually it slows, often dramatically, simulating "arrest". A hr of 20-30 with 3-5 second pauses is not uncommon. This common form of vasovagal syncope is called "cardio-inhibitory". But occasionally one sees vagal reactions with pure vasodilatation in which BP falls but heart rate doesn't.
Goes down. There are two type of vasovagal syncope 1. Vasodepressor response means blood pressure goes down 2. Cardiodepressor response means heart rate goes down. You can experience either or both responses.