I keep getting stuck on the NY Times Sudoku medium difficulty puzzles. I go through one number at a time and narrow down large box where that number can only be in two spaces. When I finish the nines I start back at the ones using the information from what numbers were determined in the first step. Then I go to the rows and columns and find ones that have around three or less boxes left, then I fill those candidates in. Then I'm stuck. Any thoughts on what strategy I'm not understanding or what things I could look for?
Stiv's answer is right (+1), but let me share another technique which might help you discover this in the future.
You've rightfully concluded that the last three columns in the second row are 1-5-8 (in no particular order). That means you can put a mark like this freehand circle around them:
and conclude the other three cells in the top right 3x3 must be filled with the other remaining numbers, 6-7-9 (again in no particular order).
6 and 7 are already out for the cell in the third row, so it must be a 9.
This trick works with pairs as well, perhaps even more often than with triples.
Sometimes you have to look at rows and columns which have more than 3 empty spaces left! Consider the space on the third row between the 6 and 2...
The row is missing 5, 7, 8 and 9 (four numbers). Rule out 7 and 8 immediately because they exist in the same column already. Then notice it cannot be a 5 because there would be no legal spaces left to place a 5 in the top-left 3x3 box. Thus this space must be a 9.
Your strategy should work for most sudokus if you don't limit yourself in this arbitrary way. Consider ALL rows and ALL columns, even if they have 4 or more empty squares, and you should find places you can make progress, like in my example above. Good luck!
Sometimes you need to look not only to separate numbers, rows and columns, but also the whole rows and columns of 3x3 blocks. In your example, you can notice that in top right 3x3 square, number 7 can be only in the top row. This cancels the possibility of number 7 being in the first row in the left block, leaving the number 9 the only option in the top left corner.
Spoiling all the candidates in the attached image, but useful tool that can help you with hints, and the strategy behind the hint: https://www.sudoku-solutions.com/index.php?page=sudoku9by9
I think it usually gives you the easiest hints first, and then more and more complex strategies as needed.
When you find rows and columns that have only three boxes left you fill those numbers in (green in the image). Switch contexts and realize you now have only three empty boxes in the top-right square.
2,3,4 are given and you know the middle row is 1,5,8 (green), so the other three boxes have to be 6,7,9. The lower-left box (red) has a 6 in the row already and a 7 in the column so it has to be 9. Using that you can write in the options for the the lower row of the top-left box and what the value in the top-left square of that box must be.
Starting with the givens in the original puzzle (not taking your eliminations into account) and after propagating the obvious constraints from the givens, the puzzle is solved with:
a hidden-single-in-a-column (==> r3c1 = 8),
followed by two interactions between rows and blocks (the most elementary rules after singles):
5 in row r9 ==> r7c2 ≠ 5
6 in row r5 ==> r4c8 ≠ 6, r4c7 ≠ 6
- followed by singles