Doctor Who has been my favorite TV show for as long as I can remember. Literally. I’ve been a fan longer than the current Doctor has been alive. I’m 34, and the show was already 15 years old when I was born. So there’s a lot there, and it can be hard finding a solid starting point.
The revived series is a direct continuation of the old. Well, they skipped some in universe time, but its the same Time Lord, it’s not a remake or reimagining or anything like that. It just picks up a few years later in The Doctors life- this has been reinforced several times when they showed his prior incarnations and they were all the classic era Doctors.
That being said, the revival did pull a major reset button in killing off all of the other Time Lords in the time period they skipped over, making it easier to get into. You could start watching with any episode and catch up on the basics fairly quickly, but for a show with as long a history and as complicated a mythology, it can help to choose a specific point to start.
The introduction of a new Doctor is a good place, you don’t have to worry about things like regeneration popping in too quickly, and you often get a new Companion who needs an infodump. For the new series, these are “Rose”- the first of the new series, with Christopher Eccleston in the lead. ”The Christmas Invasion” is David Tennants first full episode, though he briefly appeared in Ecclestons last(The Parting of the Ways). Matt Smith made his full episode debut in “The Eleventh Hour”, though again, he was briefly in Tennants last, “The End of Time part 2”.
I’d recommend “The Eleventh Hour”. He’s my favorite Doctor, and there was a bit of a shift in tone away from science fiction and towards fairy tale which has worked out wonderfully(though I was skeptical at the time). Amazing balance in his era between an overall story arc with episodes that can be enjoyed standalone.
Another good point to start would be the introduction of a new Companion. Companions are humans(usually) who travel with The Doctor. Most of the time they have little to no idea of what goes on outside Earth, so they require an infodump very early, which can help new viewers catch up- and even absent the specific information, following the companions journey from “it’s bigger on the inside” to “galactic superhero” can help get you up to speed.
Apart from “The Christmas Invasion”, the episodes above all serve for this. ”The Doctor Dances” introduces Captain Jack Harkness. ”The Christmas Bride” introduces Donna Noble, who becomes a continuing companion in “Partners in Crime”. Martha Jones comes on board in “Smith and Jones”. Rory Williams joins his then fiancee Amy on the Tardis in “Vampires of Venice”. The new Christmas special, the name of which I can’t recall at the moment, introduces yet another companion.
Some one off companion episodes of note, that can serve well for new viewers: “The Impossible Astronaut/Day of the Moon” has Mark Shepard playing a disgraced former FBI agent that helps the Tardis crew. There is an unresolved cliffhanger in the beginning, but it doesn’t impede the stories strength when viewed standalone- but it’s a hell of a payoff if you stick with the season. I’d also suggest “Dinosaurs on a Spaceship”, with Rory Williams’ dad showing up as the deal with, well, dinosaurs on a spaceship.
“Blink”, a season 3 episode, is also frequently recommended. While it doesn’t fit the above categories, it is very light on other stuff you need to know to enjoy it, and goes into the nonlinear storytelling that Moffatt loves so much. It’s not a linear story told in a nonlinear way, the story is actually fundamentally nonlinear. The Weeping Angels introduced here are an incredibly effective monster.
As for showing it to kids, it’s popular with kids but it’s also known to scare them, and the production staff delights in this- they loved being scared by it when they were kids. Just consider your own kids and how they react to scary things.