A few simple steps:
1) On the computer you want to move away from (I call it old computer from now on): Open Lr. In the Library Module / Folders panel create a new folder. Then drag-and-drop all your image folders on top of that new folder. After that this is the parent-folder for all your image folders.
If you have a single parent folder already you can skip this step.
2) Backup your catalog. Just for safety.
3) Close Lr. With Win Explorer / Mac Finder move or copy (whichever applies for you) your newly created parent folder (together with all its image sub-folders) to the new computer (or an external hard drive).
This is important!: At this point do NOT change the names of the folders or their structure.
4) Copy the Lr catalog, i.e. the file with the extension < .lrcat> to the new computer. You can also copy/move the previews folder, i.e. the folder with the extension < .lrdat> to the new computer, although this is not strictly necessary. If Lr doesn't find previews it will create new ones. Double-click the catalog-file to open Lr with the moved catalog. Naturally this requires that Lr is installed on your new computer.
5) In Library Module / Folders panel right-click the parent folder for all image folders (it should show a question mark) and select <Find Missing folder> or <Update folder loacation>.
Navigate to the location of the parent folder and select it. Lr will find all the subfolders on its own.