Since I discovered and started watching Doctor Who in 2013, it's become one of my favorite shows of all time. I have yet to finish the classic series but have watched the modern series multiple times.
The grand mix of humor, cheesiness, time travel, aliens, and emotion fills me with awe. The overall excellence in writing, acting, and storytelling has captivated me. The always-changing cast and plot points keep me on the edge of my seat.
Here is part 1 of a list of my favorite episodes of the modern series (which begins in 2005). I call the series numbers "seasons" because I'm American, in case that causes any confusion. Spoilers, of course.

Season 1 Episode 9: The Empty Child
Season 1 Episode 10: The Doctor Dances

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In this two-parter, The Doctor and Rose arrive in 1941 London in the middle of the Blitz, and meet Captain Jack Harkness and a child in a gas mask who's looking for his Mummy.
"Everybody lives, Rose. Just this once, everybody lives!"

Season 2 Episode 4: The Girl in the Fireplace

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The Doctor, Rose, and Mickey land on a spaceship in the 51st century that has various windows to France in the 18th century. They meet Madame de Pompadour and some masked clockwork androids.
"Who the hell are you?"
"I'm The Doctor, and I just snogged Madame de Pompadour!"

Season 3 Episode 8: Human Nature
Season 3 Episode 9: The Family of Blood

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In this two-parter, The Doctor and Martha land in 1913 at a school for boys. Thanks to a fob watch, The Doctor becomes a human teacher named John Smith to hide from the "Family of Blood". Martha acts as a maid and, secretly, The Doctor's protector.
"He's like fire and ice and rage. He's like the night and the storm in the heart of the sun. He's ancient and forever. He burns at the center of time and can see the turn of the universe. And...he's wonderful."

Season 3 Episode 10: Blink

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A woman named Sally Sparrow encounters the Weeping Angels; statue creatures who move only when no one is looking. If they reach you, they send you back in time to live out your life in the wrong time period. Sally has a conversation with a man called The Doctor through a DVD from the future.
"Don't blink. Don't even blink. Blink and you're dead. They are fast; faster than you can believe. Don't turn your back, don't look away, and DON'T BLINK."

Season 4 Episode 8: Silence in the Library
Season 4 Episode 9: Forest of the Dead

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In this two-parter, The Doctor and Donna visit an empty planet-sized library in the 51st century. They encounter the Vashta Nerada, who live in the shadows and feed on flesh. The Doctor meets archeologist River Song, who seems to know him very well.
"I trust that man to the end of the Universe. And actually, we've been."

Season 4 Episode 12: The Stolen Earth
Season 4 Episode 13: Journey's End

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In this two-parter, The Doctor, Donna, Martha, Rose, Mickey, Jack, & Sarah Jane (among others) meet again and team up to defeat Davros and the Daleks, who have stolen Earth and 26 other planets.
"It's gone dark...it's them aliens again."

Season 5 Episode 1: The Eleventh Hour

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The Doctor crash-lands on Earth and meets a Scottish girl called Amelia Pond, who has a crack in her wall.
"So, all of time and space, everything that ever happened or ever will. Where do you want to start?"

Season 5 Episode 10: Vincent and the Doctor

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This might be my favorite episode of all. The Doctor and Amy find a dark creature in a Van Gogh painting that wasn't there before, and travel to France in 1890 to find out what it is.
"The way I see it, every life is a pile of good things and bad things. The good things don't always soften the bad things, but vice-versa, the bad things don't necessarily spoil the good things and make them unimportant."

Season 5 Episode 12: The Pandorica Opens
Season 5 Episode 13: The Big Bang

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In this two-parter, The Doctor and Amy get a call from River Song to come to Roman Britain in 102 AD, where a legendary prison rests under Stonehenge.
"We're all stories in the end. Just make it a good one, eh?"

Thanks for reading! Check out part 2 if you'd like.