Brendan Emmett Quigley releases two free puzzles a week on his site; the New Yorker is free if you don’t let cookies accumulate; the LA Times is ad-supported for nonsubscribers, and the Wall Street Journal is free. I could be forgetting some more free ones.

And there are many subscription-based crossword constructors out there.

The NYT crossword is one of the oldest and certainly the best-known American-style crossword out there; it set many conventions (no isolated letters with no crossing word, 180 degree rotational symmetry) and with Wil Shortz’s becoming editor of the puzzle in the 1990’s changed the vocabulary of crosswords significantly by ditching old, obscure words. They even use mild curses occasionally these days (note: internet-only crosswords often do not censor their language).

Difficulty rises steadily from Monday to Saturday, with Sunday being about a Thursday difficulty but larger. Thursday and Sunday have themes; various forms of wordplay can be found in them, such as rebuses (in the lingo, this means multiple letters in one square even if it’s not an actual rebus-able set of letters), making Wednesday to Thursday one of the hardest leaps for a new solver.

A Saturday NYT is the hardest non-specialist American-style crossword out there. As such, it’s considered the baseline for serious cruciverbalists. If you can’t do them, every week, with no or minimal hints, you will struggle badly with more challenging puzzles.