I want to start my biohacking journey with using the NFC fingernails, but I don't know how to make it do things

First, read all the articles in the DT Info category… That should get you started on your understanding…

The NFC Leds don’t actually have a tag in them. They are just an LED so they cannot be used in NFC applications. They just light up when presented to an NFC reader.

If you want to just get started with playing around with NFC before you get an implant, buy some NFC tags off ebay or amazon. Just make sure they are NFC. They come in card, key fob and even sticker format. You can use your NFC enabled phone to read and write them.

Opening an electronic door requires some knowledge about what kind of lock/access control it is. There are many different types and most commercial access controls are not compatible with NFC tags. You may need a different type of tag depending on the reader. Also see Product Compatibility List

There is info on this forum about unlocking your computer with a tag.