First of all, let's define what a real crossword puzzle is.
- Each space for a letter must appear in one across answer and one down answer
- No answer shall be fewer than three letters long
Most of your better crosswords in major daily newspapers follow these rules. Any puzzle that does not should be avoided!
OK, so the first thing most people do is go through all the across answers, then all the down answers. That's fine, but the problem is that you usually end up filling in so few answers with this technique that most people give up. I think its value is mostly to get you familiar with the questions and start letting them stew around in your brain for a while.
The real way to solve a puzzle is to work from the corners, switching back and forth between across and down, working on small blocks.
Don't be afraid to try answers, but do not get so in love with an answer that you won't change it -- if an answer in the other direction makes more sense but won't fit because of letters in your answer, then your answer is wrong.
Also, look for "the key" to the puzzle. Most modern crosswords, and especially the New York Times, will have special theme answers that are typically the longest ones and all related. If you can figure out the theme, you can start really filling in some squares.
Some random tips: Watch out for clues with question marks in them -- that usually means the answer is a pun. On the New York Times, definitely be on the lookout for answers that seems very obviously correct but just won't fit the space: For example, "Old phone company," three letters, first two are MA ... the answer should be MABELL ("Ma Bell" the old AT&T), but how to make it fit? Well, what if you used MA and a little drawing of a bell in that spot? When this happens, you can be sure that it is also the puzzle theme and will appear in a lot of answers. I find this the most challenging and annoying of puzzle tricks.