A list of some of the episodes I consider exemplary, in the best sense of the word, of the Doctor Who experience, in order of their release:
A shopgirl encounters something scary after hours at work, is saved by a very strange fellow who proceeds to blow up her place of work, and has her life changed forever. Our first introduction to the new Doctor, as well as hers. Sets the tone for the series in fine fashion.
Rose wants to see her father, who died in a car crash when she was too small to know, and can't resist the chance to save him. The complications that ensue are anything but predictable, and the emotional consequences are dealt with compassionately.
"The Parting of the Ways"
Part 2 of the series (season) ending tale, with Rose and the Doctor facing the Daleks and the end of humanity... and the end of the Doctor, or at least this Doctor.
"The Christmas Invasion"
First episode of a new series, with David Tennant taking up the mantle with admirable panache. Earth encounters an aggressive alien species, and the Doctor is having trouble completing his new incarnation, leaving Rose to try to fill his shoes.
"The Girl in the Fireplace"
The Doctor comes to the aid of a frightened French girl in the 18th century while investigating a space station in the far future. Sweet and heartbreaking.
"Love and Monsters"
An episode from the point of view of someone who's been trying to find out about the Doctor since he was a child and found the Doctor standing in his kitchen one night. A fan's ultimate dream with sobering consequences, and lots of Electric Light Orchestra.
The end of the second series, with Rose and the Doctor taking on the Cybermen, Rose's dad alive in an alternate timeline, and another parting of ways.
"The Runaway Bride"
From the end of "Doomsday", as the Doctor leaves Rose behind, Donna Noble appears in the Tardis in her wedding dress, having vanished during her walk down the aisle. Catherine Tate brings a comic energy to her time with the Doctor, turning down the chance to fly off with him at the end.
"Smith and Jones"
Doctor-in-training Martha Jones meets patient John Smith at her hospital, which gets transported to the surface of the Moon by rhino-like aliens searching for a dangerous fugitive. Mr. Smith is a Doctor unlike any Martha's ever met before...
"Human Nature" and "Family of Blood"
A two-part adventure revolving around the Doctor going to ground in 1918 as a professor at an English boarding school, submerging his very being to escape detection by a body-snatching family of aliens bent on obtaining the secrets of a Timelord. Martha, taking on a position as a maid at the school, is the guardian of the Doctor's very self, having to make difficult adjustments and deal with the Doctor falling in love with the headmistress, as well as 1918 attitudes toward class and race.
Another episode that revolves around the Doctor in his absence. A young woman investigating an old house with a friend discovers a message to her from the Doctor... behind the peeling wallpaper... from 1969, warning her about Weeping Angels, and a hidden message on DVDs warning her not to look away or even blink. Then her friend goes missing...
"Utopia", "The Sound of Drums", "The Last of the Timelords"
At the far end of history, near the end of the line for mankind, the Doctor finds, or rather is found by, someone he thought was dead and gone. The consequences back in the 21st century are, to say the least, far-reaching, and Martha Jones comes to a conclusion regarding her time with the Doctor.
"Partners in Crime"
Donna Noble and the Doctor are both investigating a company making claims that seem to be too good to be true... for a very good, and very bizarre, reason. Equal parts suspense and slapstick as Donna and the Doctor take to the timestream...
"The Doctor's Daughter"
Martha, Donna and the Doctor try to sort out a war on a barren planet, and the Doctor finds himself with an instant daughter, courtesy of a tissue sample and a quick-cloning machine...
"Silence in the Library" and "Forest of the Dead"
The Doctor takes Donna to a library covering an entire planet, which is strangely deserted, and somehow linked to the dreams of a little girl. Then, to the Doctor's dismay, they meet an expedition led by archaeologist River Song, who seems to know quite a bit more than she's telling about the Doctor. And she has a sonic screwdriver. That she says he gave her...
What would have happened if Donna had never met the Doctor? And how can she get back?
"The Stolen Earth", "Journey's End"
Davros and his Daleks, bent on destroying all of space, time, and reality, are met by a determined lot: The Doctor and a group of past and present companions. Hardly a fair fight to begin with, but add a couple of Doctors to the mix, in an unusual way...
"The End of Time"
A two-part special, with the Doctor trying to overcome a prophecy of his own end, an adversary who won't stay dead, and an entire planet that shouldn't be...
"The Eleventh Hour"
The Doctor has just regenerated, blowing up some important bits of the Tardis in the progress, and crashes in the backyard of a girl who has a very scary crack in her wall. Matt Smith begins a new chapter in the life of the Doctor, with Karen Gillan as Amy Pond, a girl who has a lot to deal with in the first place...
"The Time of Angels", "Flesh and Stone"
River Song catches a ride on the TARDIS in the middle of space, leading the Doctor and Amy to a crashed ship and far too many Weeping Angels. Then the crack from Amy's wall shows up...
"Vampires of Venice"
The Doctor is determined to put Amy and Rory's wedding back on track, so he takes them to Venice for a romantic getaway. Vampires are definitely not part of the plan...
Visiting Amy and Rory in their little village while they wait for their first child, the Doctor and companions nod off... and wake up aboard the TARDIS, with an unpleasant visitor who poses the question: Which deadly situation is the real one?
"The Hungry Earth", "Cold Blood"
An excavation is losing people, and Amy becomes one, swallowed up by a hole in the ground. Then the trouble starts...
"The Pandorica Opens", "The Big Bang"
The Doctor gets a message from River Song that the mythical Pandorica, the perfect prison, is about to open. When it does, there are a stunning number of surprises, beginning with the end of the universe. And River is aboard the TARDIS as it's about to explode...
"The Impossible Astronaut", "Day of the Moon"
The Doctor gathers Amy, Rory and River Song in Utah, to plan a trip to 1969 and space. Then something really big happens. And there are aliens nobody can see...
"The Doctor's Wife"
Finding a distress message from a Timelord, the Doctor finds himself face-to-face with a woman claiming to be the TARDIS. And Amy and Rory are aboard...
"A Good Man Goes to War"
Amy's been kidnapped, her baby's being taken from her, and the one man in the universe you do not want to get in the way of is on his way: Rory. And River Song knows what's happening... and won't come to help. Or maybe she can't...
"Let's Kill Hitler"
The Doctor's still looking for Amy's daughter, her childhood friend Mels is on the lam, and they all end up in 1938 with a killer robot and Berlin full of Nazis. And suddenly, a lot of things come together...
"The Girl Who Waited"
When the Doctor takes Rory and Amy to a planet with a wide array of scenery and a deadly plague, opening the wrong door puts Amy on a different timestream from Rory and the Doctor... and means Rory has a choice to make.
Stopping for a quick visit on his way to a date with Destiny, the Doctor can't resist investigating a store that keeps losing staff, so the Doctor gets himself a new job. And River finds something unexpected waiting for her...
"The Wedding of River Song"
The season comes full circle, as does all of history. Time is out of joint, the Doctor has to put it right, and River will do whatever she can to keep that from happening... because of how it all started...