My advice is to pick a daily crossword and stick with it, going as far as you can and then reasoning the answers you didn't get the following day. This way you get to understand how the setters mind works and you dont have to wait a week or so (for prize crosswods) to get the solutions by which time you've forgotten the questions.
Don't get a book with answers at the back as you'll never learn.
Otherwise try one with cryptic and concise clues (same answer) and try not to peek. After a while you will get what the cryptic clues are pointing to