I started doing crossword puzzles in earnest a little over a year ago. I do the LA Times and the NY Times mini one every day, the Washington Post Sunday crossword each week (they have the LA Times one the rest of the week), and the three free NY Times puzzles each week.
The end-of-the-week LA Times ones are significantly easier than the NY Times ones. The LA Times ones rarely get to the Friday/Saturday NYT level. For me at least, Saturday NYT is way harder than Friday. It probably takes me one-and-a-half times as long to do Saturday over Friday.
I have noticed a huge improvement in my solving ability over time, but it took about six months before I really made a breakthrough. The first time I completed a Wednesday NYT puzzle without any help was in September. Now I can often do the end-of-the-week ones, although it may involve way too many minutes and/or putting it aside for a while.
There have been a lot of great tips already. Doing a lot of puzzles gives you a lot of “crossword-ese,” both obscure words as mentioned earlier (ENSE is a word in the Massachusetts state motto that has come up at least 3 times in the past year or so, and I’ve learned a lot about European rivers like ODER), as well as answers with a useful letter distribution that will come up over and over again (a 4-letter word for a male tennis player is almost certainly going to be ASHE; 5 letters is NADAL. I put those answers in now, even if I don’t know the actual facts involved in the clue.).
One trick that helped me enormously: I do the puzzles online, and when I feel like I’m truly stuck, I turn on the “autocheck,” which tells me which letters are wrong but doesn’t tell me what the right letters are. And then I try again until I figure it out. That’s really helped my learning, as opposed to more passively having the right answer revealed.
With the NYT specifically, a large amount of the end-of-the-week cluing is wordplay, and you have to get into that frame of mind. This is where practice really helps. There are a lot of kinds of wordplay, but it’s a finite number of types, and if you do a lot of them, you learn the types. The same types of tricks get used again. For example, “Nice summer” at the end of the week means “summer in the city of Nice in France,” or the word “summer” in French, ie, ETE. Completely fooled me the first time, but it’s come up again, and now I know.
Within the past month or so, I’ve started doing the NYT Spelling Bee game, and I’m much better at it than I used to be. I think that staring at half-filled words has helped me with that.