Firstly, adding the game installation folder, to your security software (anti-virus, firewall, etc.) exceptions list will be required. If DirectPlay is also not installed, then you will have to install DirectPlay. This article should be helpful:
If you would like to play the game in a LAN environment, simply start the game as usual, and select the Multiplayer from game's options. Here you can host or join LAN games.
You will have to type the local IP address of the computer you want to join, within a local area network.
Host can easily check the local IP address directly from Windows. Simply press WinKey+R to open Run window and type CMD. Press Ok. In Command Prompt type IPCONFIG, and press Enter. IPv4 Address is the local IP address.
If you would like to play the game over the Internet, then you will have to use QuakeWorld. If you are using Galaxy select the game in Galaxy -> More -> QuakeWorld (client). If you are not using Galaxy, press Start Menu -> Programs -> Quake - The Offering [GOG.com] -> QuakeWorld (client).
In order to join the game you will have to know the IP address of a dedicated server from QuakeWorld servers you wish to connect to.