There's a great many answers to this question, I'd say. Here are the most obvious ones explained, so you can make an informed choice:
- You could start when the show came back in its modern form, which is 2005. This would be Series 1, Episode 1, "Rose". This is in the same continuity as the classic show that dates back to 1963, but this episode marked the show's return after over a decade off-air, and it acts as a sort of 'reset' for the continuity. Then, over the years, classic & new continuity/lore seeps back in gradually over time. The problem: the first year of the show was rough in terms of effects, production and sometimes tone even in 2005, so 15 years later it isn't aged brilliantly in this regard. The writing is as sharp as it ever was, though.
- The show found its footing a little bit more when the next Doctor arrives, which is 2006. This'd be Series 2 Episode 0 (a Christmas Day special), "The Christmas Invasion". By this point the 13 episodes before this had helped them to find their feet, and the show has a tone, style and direction that'd continue to grow and get finessed without major adjustment for the next ten years.
- In 2010, the show underwent a major change - a new Doctor, a new production team, a new showrunner - new everything, basically. Tonally it follows on exactly from 05-09, however. It's the same show, just done with more money and more slickly. This is when the show goes HD, starts to look a lot less cheap (with some nasty CGI here and there), and also when the show with a slight tonal change built up a particularly big cult adult following in the US (in the UK, Doctor Who is more of a family show that happens to have a cult nerd following - think Harry Potter, basically). This is also a great jumping-on point and is hands down the best-quality 'starting' episode in this list, though it also features lore, characters, etc established over the previous five years. The first episode here is "The Eleventh Hour," Series 5 episode 1.
- While there's a Doctor change in between the third option and this one it isn't nearly as friendly for newcomers, so - the final option is the most recent, and is basically the same as the previous one - a 'reboot' of the show that's a great jumping-on point. This introduces the current Doctor and main cast, and is written by the current head writer. This is Series 11, Episode 1, "The Woman Who Fell to Earth". This is one season (15 episodes) behind this week's aired episode, fwiw.
Those are the options. Personally I love the idea of starting at "Rose", but that means 296 episodes(!) across 11 series'. Some people will advise you start with specific episodes, or watch certain episodes of a series and skip others... which is valid, and works because Doctor Who largely has a 'monster of the week' continuity-light structure episode-to-episode (ala Star Trek), but I personally don't recommend that - I'd pick a starting point, go in with an open mind and stick with it - give it a series or two to grow on you.
For my money the "golden age" of recent Doctor Who is probably series 4 and series 5, so two series' that bridge two of the eras highlighted above.
FWIW I wouldn't bother with old Doctor Who unless you have a real taste for really old sci-fi like Star Trek TOS - or unless the modern show grabs you. If anything from 2005-on grabs you and you become a fan, you can then dive into the world of the old iteration of the show from the 60s, 70s and 80s, which will in turn reward you with loads of lore and stuff that'll enhance your enjoyment of the modern stuff.
Anyway. That's it. Those are your options. Do it! The show is great.
