What works for me is to solve the top row (or left-most column) and then leaving those pieces untouched from that point onwards. Once the puzzle is reduced to a 2x2, you just need to shift those last 3 pieces a few times until the puzzle is all solved.

I'm not sure if you are doing the 8 puzzle (pieces numbered 1-8 in a 3x3 grid) or the 15 puzzle (pieces numbered 1-15 in 4x4 grid). For the below explanation, I'll assume you're doing the larger puzzle.

to solve the top row: ignore everything but the "1", "2", "3", and "4" tiles.
Push the "1" to the upper-left corner by just moving it left and up.
Then push the "2" to the second column, and then push it up.
Push the "3" and "4" to the right column (making sure the "3" is above the "4"). Then push them, as a pair, to the top of the column. Clear the space to the left of the "3", and push the "3" into that space, and the "4" up to the space just cleared. You'll be done with the top row.

Now, never move any of those pieces again - you'll only deal with the bottom 3 rows.

The left-most column is "1", "5", "9", "13" (with the "1" already placed). Move the "5" to the correct row, then push it left until it's in place.

Then push the "13" into the row where the "9" should go, and the "9" just right of the "13" -- move them both left to the edge, then move the "13" down and the "9" left. You'll be done with the first column, reducing the puzzle to a 3x3 puzzle (the "8 puzzle"), even though the numbers are 6,7,8 / 10,11,12 / 14,15. Solve either the 5-6-7-8 row or the 2-6-10-14 column, whichever is easier, followed by the 2-6-10-14 column (just the last two pieces need to be put into position) or the 5-6-7-8 row, respectively.

You're left with just the 11, 12, and 15 pieces in the lower-right-hand corner, which you just need to shift around until all the pieces are in place.

Help with the smaller puzzle - stuck - please