It might be helpful to clarify some terms, as it isn't always clear just from undirected reading of the documentation
Lightning Components is a broad the used to define the components that make up the "Lightning" UX that Salesforce introduced as the replacement for what is now known as the Classic UX
The Aura framework was the original framework that Salesforce built to deliver these Lightning Components when the Lightning UX was introduced. Aura was always the name of this underlying framework, but at the time, the components built using this framework were referred to as Lightning Components
Fast-forward a few years, and Salesforce made the decision to adopt a less heavy, standards-based approach to building Lightning Components, and introduced Lightning Web Components, based on the Web Component standard.
Colloquially, both of these are considered Lightning Components, but to help reduce confusion, aura-based components are now referred to as Aura Components, and LWCs as Lightning Web Components.
The transition to LWC is still underway, and the Lightning UI still relies on the Aura framework. You'll find there are many functions that Aura Components can do that LWC can't. And each release you'll find more and more enhancements to LWC.
It is my opinion that you should focus and default to LWC, but that you should complement your study with learning at least the basics of Aura and the ways that Aura can and can't coexist with LWC, for both supporting existing work, and because you will find scenarios where you'll need to use Aura as a wrapper in order to deliver your work.
Edit: to answer your last question, yes the Lightning Design System is supported in both Aura and LWC (although styling hooks are exclusive to LWC)