Twenty years ago, I would have said "Just keep doing puzzles", but with the advent of crossword-related blogs things have changed. I would suggest looking at Big Dave's site (
http://www.bigdave44.com) and fifteensquared (
http://www.fifteensquared.net), where you will be able to see the range of puzzles which are blogged by experienced solvers and then commented on by others of varying abilities. From this range (Telegraph, Guardian, Indy etc) pick a non-prize puzzle (the publication of blogs for prize crosswords is delayed until after the closing date), and attempt it.
Don't consider that you have solved a clue until you have an answer
and can explain why it is the answer. When you come to a standstill (or finish the puzzle to your satisfaction!), go to the relevant blog and read through it carefully, looking at all the explanations to check whether they match your own. Once your solving has got to the level where you feel confident enough to leave comments on the blog, you have made progress; when you reach the point where you start correcting the blogger, then you've
really made progress.