No one should be heading anywhere, the easiest way to contain a zombie outbreak would be if everyone secured their own homes and waited. That way the army and police could institute a "shoot everything that moves" policy and maneuver effectively through the area.
Most people would have enough food for a couple of days, if not you won't starve to death immediately. While there is still water, everyone should fill buckets and baths. While some people might be killed by zombies, the majority could sit it out in safety.
But if people panic, some wanting to flee, others to gather guns and food, the roads will become blocked and people might kill each other accidentally. The authorities will have a hard time keeping everything under control.
So I would secure my uni residence building with the help of other students and wait a couple of days, before trying to make an escape back to my hometown, where my family and my gun is.