Torchwood, as an institution, has always known about The Doctor. It was started with the explicit goal of protecting Great Britain from him, and other alien threats. As such, one would assume that the Torchwood Three (the Cardiff branch featured in the series), at least up through the battle at Canary Wharf where Torchwood One was destroyed, knew all about The Doctor.
As far as the staff of Torchwood Three during the course of the series:
- Jack, of course, had known about The Doctor since his introduction in the Doctor Who episode "The Empty Child", long before Torchwood
- As a Torchwood member, Tosh (when she appeared in the Doctor Who episode "Aliens of London") has met him, and would likely have known who he was. Even if she didn't recognize him immediately, his knowledge of alien biology should have tipped her off fairly quickly as to his identity.
- Tosh claims she was "covering for Owen" at that time of that episode, meaning both of them worked for Torchwood prior to Canary Wharf, and thus Owen should also have known of The Doctor, though as far as we know they never met.
- As best as I can determine, neither Gwen nor Ianto knew anything specific about The Doctor prior to "The Stolen Earth". Jack mentioned him (usually cryptically) a few times, but otherwise there was no recognition for either of them when Ten contacted Torchwood during that episode.
After the destruction of Torchwood One, Captain Jack took control of Torchwood Three and attempted to change its overall attitude about aliens. It's not clear how much of the old Torchwood information about The Doctor would have remained, though it would be entirely in character for Jack to erase as much of it as possible. That would explain how Ianto, who spent a lot of time digging through the computer archives, could remain unaware of him.