Doctor Who just works. However you watch it, as a fan or casual viewer, there is something there for you; and if there's not, well, try a different era. It helps that it's got 52 years currently under its belt, and so there is and has been for a long time, an element of nostalgia to the show-- recurring villains, references, companions or places/planets that get revisited just to please the people who've been watching long enough. But that's not all there is to it: because every year, there's some kind of hidden gem of an episode that's a shining example of great television, along with the scary, funny, tense episodes we have all come to expect from this show. One of its strongest merits is its constant adaptability. There are different writers almost every week, different companions every other series, different doctors, different locations, directors, genres, threats and ideas. For every one abysmal episode (and there are a few of them), there are some absolutely stunning ones too. I'd recommend Heaven Sent, Bad Wolf/Parting of the Ways, The Empty Child/The Doctor Dances, Blink, The Pandorica Opens, A Good Man Goes to War, Asylum of the Daleks, Flatline, and The Day of the Doctor. It's a show that never dies. Of course, it will get cancelled at some point, maybe, just as it did before; and then it will live on. It will get picked up again. TV just isn't the same without it.
If you're new, it's best to start with some classic stand-alone stories to get into them. Maybe try a few from each series to work out who your favourite Doctor/companion combination are. 'Smith and Jones' is a lovely episode to start with (it's where I started)--the season 3 opener, with a new, companion, a reintroduction to the Tenth Doctor, and a wholly entertaining episode. Other great places to start are Rose (although there's a lot of catching up to do), The Eleventh Hour (a completely brand new start-- perfect if you know absolutely nothing about anything in the show), and Deep Breath (an introduction to the current Doctor, with a few entertaining characters who have already been in the show before). Generally, starting with a Series 1-4 episode will be much easier, with simpler stories, a new companion/Doctor each series, and some enjoyable, if upsetting, season finales. Series 6-9 are harder to start at, with characters carried over from previous seasons, and plot lines and mysteries also carried on with. The individual episodes within the seasons, however, need no foreknowledge at all: for Season 6, be sure to try The Doctor's Wife and The Girl Who Waited; Season 7, try Asylum of the Daleks, Dinosaurs on a Spaceship and The Bells of St John; Season 8, try Flatline, Listen, or Kill the Moon; and Season 9, try The Girl Who Died/The Woman Who Lived, The Zygon Invasion/Inversion; and Heaven Sent (which is absolutely incredible). It's a lot of episodes, which for some seems too much. For me, however, it's never enough.
If you're new, it's best to start with some classic stand-alone stories to get into them. Maybe try a few from each series to work out who your favourite Doctor/companion combination are. 'Smith and Jones' is a lovely episode to start with (it's where I started)--the season 3 opener, with a new, companion, a reintroduction to the Tenth Doctor, and a wholly entertaining episode. Other great places to start are Rose (although there's a lot of catching up to do), The Eleventh Hour (a completely brand new start-- perfect if you know absolutely nothing about anything in the show), and Deep Breath (an introduction to the current Doctor, with a few entertaining characters who have already been in the show before). Generally, starting with a Series 1-4 episode will be much easier, with simpler stories, a new companion/Doctor each series, and some enjoyable, if upsetting, season finales. Series 6-9 are harder to start at, with characters carried over from previous seasons, and plot lines and mysteries also carried on with. The individual episodes within the seasons, however, need no foreknowledge at all: for Season 6, be sure to try The Doctor's Wife and The Girl Who Waited; Season 7, try Asylum of the Daleks, Dinosaurs on a Spaceship and The Bells of St John; Season 8, try Flatline, Listen, or Kill the Moon; and Season 9, try The Girl Who Died/The Woman Who Lived, The Zygon Invasion/Inversion; and Heaven Sent (which is absolutely incredible). It's a lot of episodes, which for some seems too much. For me, however, it's never enough.