Probably an Nvidia capture card would suit fine or the Delgato capture card too. You could also use the Xbox Game Bar in the mean time just until you get it setup. That way you at least have something to use before you buy it.
A free program that you could use is called OBS. It's excellent and very customizable, would be perfect for your needs. You also can stream with it, if you would wish to. I use it myself and never had any problems with it, high quality picture and easy to use.