As a Border Northerner living in the Southwest, I run into this all the time. We even have different names for certain things. Additionally, since I love British Literature, we read a lot that introduces my little girl to other ways to phrase (and even spell things) all the time.
I've met kids with parents from the UK and New Zealand and the kids sound like red-blooded Americans. I'm afraid there isn't a lot to be done for the most part. However, since your little one is two, if there is any particular phrase that really bothers you, I'd suggest simply treating it as you would any other grammar mistake he would make. Do you mean, "say it the correct way here?"
As he gets older, talk to him about it. Alice has had no trouble understanding that different people speak in different ways and have different names for things. She actually starts sounding a bit more like an islander (where I am from) during the summers when we are up there for a couple of months, and then reverts back when we return south in the fall - I really don't think it is even anything she is conscious of doing.
When it comes to proper speaking over-all, I wouldn't personally become too concerned unless you are in an area where the dialect is so strong it would be difficult for others to understand him if he were to travel. There is a time and place for casual language with friends too. There are times when proper speech is important and you can work with him on sorting this out the same way you would teach the difference between say, texting messages and formal reports as he grows and is more ready for those sorts of intricacies. Kids are better at getting all that sorted out than parents often give them credit.
I once had a mother concerned over her son's spoken English because she used broken English herself. He spoke with her at home in their home language or at her level of English and then came to school (in Middle School) and had grammar just as good as any of his peers. She'd never heard him speak outside home and wasn't aware of how well he did speak.
As long as he knows what the "correct" usage is by the time he is leaving for college and when to use it, he will probably be just fine.