The opening episode of the 2005 revival. I grew up when it was off-air, but I came to
Doctor Who long before the revival with re-runs of the "classic" years, and then investigated further with VHS/DVD releases and the ancillary media. I find it hard to imagine would it would be like to come to the older stories from the revival.
The revival is a continuation of the original run, albeit with a significant in-narrative time jump, and was designed to be appreciated as a new show without any baggage, with any necessary backstory being filled in over time. Once the series was re-established however, the production team succumbed to the temptation to start including nostalgic elements, and as we got closer to the 50th anniversary the gap was closed until it's now virtually become one whole series again (with a 15-year break between two seasons). There are certainly elements that tie into things from many years past; it's not absolutely essential to know the ins-and-outs of the first 26 year run but you'll get a lot more out of the modern run with at least some basic knowledge of it.
Personally, I'm wont to recommend starting with at least the original first episode, 1963's "An Unearthly Child," which I think is vastly superior to "Rose," relative to the time it was made, and will put certain enduring core elements of the series' mythology into their original context. That said, it depends much on the individual viewer's sensibility. Much of early
Who was made against the clock on a shoestring budget, is largely studio-bound, stagey, and much of its effects work is embarrassingly B-movie (and the first 6 years are in black and white), but if you can look past that and see the wildly inventive premise and stories, the wit and the charm of the whole thing then definitely start at the beginning. Its format may take some getting used to as well; 25 minute episodes, multi-part serials, none of which were designed to be watched all together in omnibus form on DVD 30/40/50 years later, so when watched in this way the structure may seem odd.
It's worth noting that
Doctor Who can be subdivided into many different eras in terms of tone and approach, as it would subtly reinvent itself every few years when a new producer and new writers would come in with new ideas. The series was initially designed as an intelligent children's semi-educational programme (which could be appreciated by parents too), and then progressed to a more action-adventure formula, became an Earthbound, James Bond-esque show in the early 70s, then took on a Gothic Horror-style flavour and so on. It largely grew up with its audience, and then descended into being very child-orientated again (to put it kindly). Nowadays it's run by fans (a bit like
The Simpsons and
Star Trek, it's been around for so long that its writers grew up with it), and you can see the influence of all eras of its history - it's a bit like the lunatics running the asylum!