The fastest way to improve your copy? There isn't one.
Read a lot, write a lot, and pay attention to what works.
Read a lot. And read better.
If you solely read ads all day long, you will be adept in the language of advertisers. You will know how they talk and the terms they use and the trendy phrases of the moment. If you're writing copy for an audience of advertisers, this works wonderfully. However, you're probably not writing ads for advertisers.
Copywriting is the practice of getting an audience to read something and then do something. How do people read? It helps to see how they write.
Sure, read the classics. Also, read the trash for sale in the book/magazine aisle at the grocery store. Pick up magazines you would never subscribe to and read them while you're on the can. Notice the simple language, the shoddy sentence structure, and how they'll use bold lettering instead of elaborate word choices to get and keep attention.
Your copy doesn't need to be eloquent, literary, or high-brown - it just needs to get the message across to those you want to share it with the most.
Every writer is a reader first. Read extensively, across all mediums and genres. Read deeply with a pencil in hand to mark down anything that jumps out at you. What draws your eye? What did you pass over? There is just as much, if not more, to be learned from reading bad writing as there is learning from the best writers.
What Worked On You?
Advertisers save ads that get their attention, but there is a big difference between an ad you like and one that works.
Consider the last thing you purchased that wasn't a grocery item - what drew you to buy that precise item?
You may not even remember the last ad you saw for the product, it might be worth looking up. Consider the headline and the copy or the images. You are clearly the target audience; what in the copy spoke to you, if anything at all?
Pulling ads from magazines and off TV commercials is a good way to get the juices flowing when it comes time to write your own copy. In the meantime, try rewriting someone else's work. Take a bit of widely distributed campaign copy and rewrite it to see if you can make it better. Chances are, what you come up with will be exponentially better than whatever an agency with red-tape is beholden to.
Think about the audience they were targeting - can you hit the same audience? How much of the copy would need to change before it is something YOU want to buy?