How does the process of installing a wireless adapter to a desktop PC go exactly?

You plug it into a USB port & it should load the drivers all by itself. Hence, plug & go.

Buy the adapter, install the adapter and software.

1)Buy a USB wireless adapter such as any of these
https://www.amazon.com/s?ie=UTF8&field-keywords=usb%20wireless%20adapter&index=blended&link_code=qs&sourceid=Mozilla-search&tag=mozilla-20
2)Plug it in, if the drivers are not installed automatically your may need a driver disk supplied.

or if you want an internal adapter then open the case, buy an adapter such as this
https://www.amazon.com/s?ie=UTF8&field-keywords=pci-e%20wireless%20adapter&index=blended&link_code=qs&sourceid=Mozilla-search&tag=mozilla-20
plug it into a free PCI-e slot. Power up and follow from 2) above