There are many very good answers about HOW to use Lightroom to do what your question asks. I thought I would provide an answer as to WHY you would want to use Lightroom to do this: Lightroom can automate creation of folders, and provide more power and flexibility in managing those images than can be achieved with folder names alone.
Lightroom is defined by its Catalog. The Catalog contains information on the images that LR manages, such as file names, locations, edits, metadata etc. It does not, however, contain any photos, only a pointer to where the folder is located. Due to this, Lightroom does not care where the photos are located, nor does it need any specific folder names........and neither should you.
The real reason for using Lightroom for moving and cataloging images is to use the automation provided by Lightroom, where it can import the images from other drives, and move them to a new drive using a folder naming methodology available within the Import function. There are several, from folders by date, sequential, sequential names etc. But Lightroom can't automatically create folder names by location, or setting or client.
If you require folder names following some personal naming convention where there is no available metadata in the file, Lightroom is going to be a tedious tool to do what you want. If instead you let some form of metadata determine the folder name, Lightroom will be fast, simple and easy to achieve your goals.
Example: if you want to name folders by photo taken date, Lightroom can easily automate this:
If you want your folders like this:
you will have significant problems with Lightroom, as you will have to define folders for every import, and every file.
So, allowing Lightroom to manage completely the naming of folders is a key to successful use of Lightroom.
But, you may worry that you currently use Folders to help you find photos, and and manage across clients and work. Indeed, folder names can be useful for this, but tedious to keep up with.
Lightroom allows you to expand on the info that folder names provided, by something called Collections and Smart Collections. Essentially, a Collection is a virtual group of photos, that can easily span across folders and even drives. Photos can also belong in more than one Collection. So you can have Collections such as:
Smith Rehearsal Dinner
Smith Wedding Ceremony
with photos in each. Photos in 'Smith Wedding Ceremony" are also in 'Smith Wedding".
You can also create Smart Collections, which are Collections that are created by something in the metadata. For example, when you import the photos from the Smith Wedding ceremony, you might add tags such as Smith, Wedding, Ceremony. The Smart Collection will be defined as those images that have all three tags, plus have the Pick flag. This way, your folders and your collections are all automatic...all you have to do is edit! (use the Pick flag to quickly pick out your preferred images)
So, Lightroom is ideally designed to do what you ask, and remove complexity as well as time from managing your photos. The key, however, is allowing Lightroom to manage the images for you, the way it was designed.
note: the downside to leveraging Lightroom is this way is if you need
that metadata that the folder names supply to you when not using
Lightroom. By doing things Lightroom's way, you make managing images
much simpler, but you do lose the fidelity of foldernames OUTSIDE of