Hello, Elle!
No, I mean the crossword in the main Times paper, not the quick cryptic one.
The Guardian have all their past crosswords archived on line. You can access them free. I usually print some off now and again. If you are improving, Elle, I would suggest you start off with Rufus. He is normally in on a Monday. I think he is the easiest setter. They also archive their crosswords by setter, too.
So, If you liked Brummie, say, click on Brummie, and all his crosswords come up back to 1997, or whenever he first started. I am currently doing Arachne's from 2003!