Piracy: ꜱᴀɪʟ ᴛʜᴇ ʜɪɢʜ ꜱᴇᴀꜱ

Short answer: I'd probably start with TVC for video stuff and MAM for audio stuff. They definitely won't get you all the way there, but they'd be a decent enough start for a collection and ratio isn't hard on either one.

Longer answer: The main series itself and the theatrical movies shouldn't be too hard, and even the main spinoffs (K9 and Company, Torchwood, The Sarah Jane Adventures, even Class) shouldn't be too bad. I think where you're going to run into issues are things like the fan reconstructions of lost episodes, the various animated stories, short stuff like the Tardisodes, there's the behind the scenes programmes (Doctor Who Confidential, Totally Doctor Who, Doctor Who Extra, etc), the Red Nose specials, etc.

Then you get into the other stuff - official BBC stuff but not-on-video, all the fan stuff made during the lost years that had official / semi-official / unofficial-but-winked-at licensing, etc: There have been official BBC audio plays and radio dramas. I think most of Big Finish audios have leaked, though collecting and keeping up with them would be a chore - it's a pretty big collection, and they still make multiple stories every month. The ReelTime and BBV videos can probably be tracked down with a bit of effort (I do like the Myth Makers series). I don't know if any of the theatrical productions have been professionally filmed, though there's likely amateur video (but watching an audience video is often tedious and I'm not sure how much you want to get into this). Etc, etc.

If you want to get even further into things, there are novels and comics and annuals and all sorts of other stuff. I guess the question is, where do you want to start, and how far do you want to go?

Again, personally, I would start with TVC and MAM and see how far I wanted to go. I mean, the show has changed a lot over the years, some of it's missing, it's in various formats, stuff in different formats us considered canonical or non-canonical depending on the person. Even if you're willing to sit through black-and-white video recordings from the William Hartnell years, do you want to sit through fan reconstructions of ten of the twelve episodes of "The Daleks' Master Plan"? Do you want to read Lungbarrow in preparation for the TV movie? It's a long, complex, and time-consuming continuity.