Let's suppose you have the basics down, which means you can solve the Friday puzzle in USA Today every time. Now, what?
IMO, the next thing you should master is the idea of two strong links and the Turbot Fish patterns. That includes X-wings, Skyscrapers, Kites, and variations of Empty Rectangles, etc. If you approach it from the point of view of strong links, there is a laundry list of named techniques you won't need to bother about, including Swordfish.
That said, it depends on where your puzzles come from. Most puzzle publishers screen their puzzles to be solved by certain techniques. The most usual is an X-wing (UK) or XY-wing (USA).
If you want some fun, try Menneske. His puzzles are not graded by specific techniques required. Try his Very Hard (0) puzzles. I find them quite enjoyable.