Start with What You Know Right Away
A popular strategy for test-taking is to go through all the questions and skip over the ones you don't know. And even though they aren't multiple-choice, you can do the same with crossword puzzles -- thankfully, without being graded at the end. Don't spend too much time tearing your hair out over a clue you just can't crack. Go on to the easy ones, because that strategy comes with a bonus: filling in the answers you know will provide letters for the ones you don't. That narrows down the possibilities significantly, especially for clues where more than one answer could apply.