Just drag the complete hierarchy of folders to the network drive using your operating system.
Now when you will open Lightroom it will show you question mark on the folders as folders would be missing from original location.
Right click on the top folder and then choose find missing folder and point that to the folder that is placed at network drive.
That is all you need to do.
Please do not change the hierarchy of folders or do not rename folders/files at the time of this operation.
Is just dragging folders within LR to the network drive in the catalog from the local drive best option ?
No, it is a poor choice. There are rare but persistent reports that the photos being moved simply vanish, and cannot be found either in Lightroom or by the operating system. Re: Moved files within Lightroom, 2/3 flat out disappeared?!
The best and safest option is to COPY (not move) the photos (the entire hierarchy of folders) to your network drive using your operating system; then in Lightroom, right-click on the top level folder of this moved hierarchy and select Update Folder Location. Point to the new location of this top level folder on the network drive. Then once you are 100% sure you have completed the move properly and everything is working in Lightroom, you can use your operating system to delete the files from your local drive. More details and pictures are shown here (scroll down to the section entitled "Updating Folder Location"): Adobe Lightroom - Find moved or missing files and folders