In 2019 I bought a second property to use as my primary residence. I spent a few months renovating and moved into the property in December 2019. I paid points as part of my mortgage for this second property. My understanding is that I do not need to amortize those points, because I'm using this property as my primary residence (even though it's my second time purchasing a home). Is that correct?
Also, not sure if this is relevant, but I started listing my first property for rent in December, but it didn't get rented out in 2019, so I won't be filing a Schedule E as I had no rental income in 2019.
It depends.
Typically you would amortize the points over the length of the loan. Attached is a flowchart from the IRS that you can use to determine if you can expense the points this year or if you will need to amortize them based upon your specific set of circumstances.
As far as your rental property, you are correct. Since it did not get rented out in 2019, you do not have any filing requirements for the property.