Upgrade to HD DVR - DirecTV

I currently have 2 D12-100 standard definition receivers connected to non 4K TV's. I also have WiFi that has limited monthly data.
If I change to HD with DVR, will I need to connect to the internet?
Also what equipment will I need for the 2 different tv's, and be able to record different shows?

This question, "Upgrade to HD DVR," is about DirecTV channels, programming, equipment, and sports.