We put in an offer on a house in December 2017, and closed on the house at the beginning of January 2018. Would I put I purchased the house on this years taxes, or next?

You only became the owner of the house in January 2018. Only indicate that you purchase on your 2018 tax return (next year)

And you should claim the deductions related to the house such as mortgage interest and property taxes on your 2018 tax return.