Mobile
For the best experience, keep the following shortcuts in mind.
Previous/Next - Pressing the previous or next buttons on the left or right side of the current clue respectively will go to the previous or next logical entry.
Clicking Current Clue - When a cell at the intersection of two entries is focused, pressing the Current Clue will switch between the across and down entries that the cell is part of. This can also be achieved by manually clicking the cell at an intersection.
All Clues - You can see clues by clicking the All Clues link at the top of the crossword. Clicking any clue in this view will automatically focus on that clue and open the crossword again.
Desktop
The following few but powerful shortcuts will enable you to solve a crossword puzzle without ever needing to touch the mouse.
Left/Right keys - navigate from one cell to another.
Alt Key - When a cell at the intersection of two entries is focused, pressing the Alt key will switch between the across and down entries that the cell is part of. This can also be achieved by manually clicking the cell at an intersection.
F1/F2 - Go to previous or next logical entry.