There are a couple of ways to save Geometry Node groups.
As you can see in the answers linked in the comments already, you can use Asset Browser for it. You should look into how the Asset Browser works in general first. Short version is: you mark Blender's data blocks as assets, and save them to some files in some folder that you set up as your asset library in Blender's preferences and then Blender loads those assets from those files in the folder as your asset library in the Asset Browser. This works for all sorts of stuff in Blender, node trees/groups including. This is probably the best way to do it and you should look into that.
I wanted to mention another way for very specific situation. If you want to save some things that you use all the time in pretty much all kinds of projects you do, you can save Blender's [data blocks](data blocks are objects, meshes, materials, textures, particle systems, material/compositing/GN node groups and pretty much all and any kind of data Blender saves in a file) to the default startup file, so they are available when you open Blender with the default scene. In order to do so, you need to open Blender with it's default startup scene and append everything you want from other files or make it there and make sure all the stuff that is not yet used has "Fake User". It's usually the shield icon and you can find it all over the UI, but in case of Geometry Nodes it's here:
There are other ways of setting Fake User like in the Outliner:
Once you have everything you want in the file and also everything you don't want removed you can save it as default Startup file:
Keep in mind that everything is saved even things like UI and tool settings and you will always have that file loaded when you start Blender so it makes sense to make it convenient for what you do with Blender.