A subscription always includes the latest full version, upgraded as they come at no extra cost.
Other than that it works as it always has. The software is installed on your computer and everything works locally. You save to your local hard drive - unless you choose the option to save in the cloud.
For anyone working with largish camera files and managing a large archive, the "cloud" is still completely unrealistic. Internet bandwidth is nowhere near able to cope with a high volume workflow. And as bandwidth increases, so do average file sizes. It never catches up.
All that said, a laptop isn't ideal for Photoshop. At least make sure it's one with a lot of disk space (Photoshop eats disk space for breakfast) and good specs overall.