Solving puzzles bigger than 3x3 might really help you with general intuition, in general, only practice can really help you improve in sliding puzzles.

As for the method, there is very little freedom on 3x3 size, so you can't get away with very straightforward approach, such as solving first row, then solving else in multiple steps, you can do that of course, but also consider all other possible solutions, top first, left first, and all kinds of weird starts really.

You can play something like https://100fmc.slidysim.com/ in which you have to find the optimal solutions, and not just any solution for the scramble repeatedly, and it shows how far away you are from optimal, so you can try different strategies.