Why do you need to reach Legend quickly? Also, there are no prizes in the Legend Club currently, so it really is more for you and your sense of accomplishment than it is anything else.
First thing is to familiarize yourself with the points, so that you can plan what you want to use for gaining league points: https://www.chess.com/leagues Arena tournaments are probably best for grinding because you get the regular game points AND you also might get bonus arena points (win streaks and such like that).
When I was working my way into the Legend Club, I mostly cared about my rapid rating and didn't care for my bullet or blitz, so I mostly played 3 min blitz arenas for league points and sometimes played 1 min bullet arenas. I just played rapid normally during this time and if I got any points for this, then that is just a little "extra" but wasn't my focus.
Like most people, I also stopped caring about the Leagues for a while because the amount you need to play to advance was sometimes too crazy and would cause my ratings to crash too much just to keep playing even when not in the best form. I just got to the point where I figured it wasn't worth it and then I decided to try again months later when I was able to advance the last few divisions I hadn't advanced past before.
I need to get to legend league the quickest way.