Work in Clusters of Columns
Crossword puzzles are a little like spiderwebs; each answer is connected to the others nearby to form a woven network of letters. Tugging on one string of letters affects all the others, too. Most crossword puzzles are naturally divided into several groups of up and down clues, connected to the other groups by longer answers. With that in mind, working through the puzzle one cluster of columns at a time -- as opposed to going through all of the across clues at once, followed by all of the down clues -- is a good strategy. Each word you fill in will help solve the others around it.