The Holme, Coln, Calder, Dearne, Don are all close by for Trout and Grayling fishing though some also contain course as well. I know that Brownies have been caught on the fly right outside Sheffield Wednesday's ground in the middle of that city. Salmon and Sea Trout we don't have that much unfortunately though things are getting better. I think the Yorkshire Esk on the North East coast is one of our jewels. Also the River Ribble though a lot of that is unfortunately in Lancashire

If you're tempted to commit treason though you can also try the Lune for migratories.
You do get some Salmon and Sea Trout going up the Ure/Nidd/Swale. I also hear of them in the Wharf and extremely rarely in the Aire. I believe a Sea trout was recently caught in the Calder near Wakefield! Migratories on these rivers all seems to be kept very hush hush though
I have heard that they have stocked a lot of young Salmon fry into the Trent a few years ago in the hope of reviving that river back to being the Pristine Salmon river it was in Victorian times. Read more here:
http://www.derbyshirebiodiversity.or..._the_Trent.pdf
Cheers,
John