Personally, I would say "to" but I think this varies by region. Where I'm from, we would never say go "into" town. However, I have heard people word it that way before.
An example:
If you live in California and you are going to Wisconsin (a different state) you might say "I'm leaving tomorrow to go to Wisconsin."
Whereas, if you live in Ohio in a nearby town and were going to Cincinnati one of the cities, then you might say I'm going into the city tomorrow meaning you live nearby but not in the city.
Either way, I would still say "to" but this is just one example of the context I have heard "into" used in.
Nonetheless, I'm not sure which is "grammatically" correct, maybe someone can answer that. But, as a native speaker myself, I think any native speaker would know what you meant regardless of which one you use. I have heard both, but almost always it has been "to".