I would have a hard time choosing the 'best' episodes; I have seen a few of the older ones, but not enough to select my opinion of the cr?me de la cr?me. So instead here are a few recent favorites, starting with the two most recent, then in random order: 'Time Heist' (just aired this week, in fact, and had a decent review posted of it here:
http://www.avclub.com/tvclub/doctor-who-time-heist-209477), 'The Doctor's Wife,' 'Blink,' 'The Girl in the Fireplace,' 'Silence in the Library,' 'Human Nature,' 'The Family of Blood,' 'Utopia,' and 'Midnight.'
And you can watch many of these on Netflix, or otherwise online.
'Blink' would be a particularly good start since it is a strong 'stand-alone' episode; i.e., a decent episode that could have been from any series about time travel, and the viewer does not need to know anything about
Doctor Who. There are these weeping angel statues that are actually carnivorous lifeforms. To get rid of them, the main protagonist for the episode picks up clues left behind by a stranded time traveler.
Short descriptions:
Time Heist. A classic bank heist plot with some time travel thrown in. The Doctor, our time traveling main protagonist, wants to break into the most secure bank in human history, which requires everyone on his team (including himself) having their memory wiped up to the point they enter the bank. Turns out there is a humanitarian reason for infiltrating the bank. Also, the main antagonist feeds on your thoughts: the more you think, the easier he feeds.
The Doctor's Wife. The Doctor is lured to an small planet by a malicious entity who wants his time machine. On this planet, he meets a most charismatic and eccentric woman who he actually knows quite well, and who is the answer to him escaping the grasp of his host.
The Girl in the Fireplace. The Doctor happens upon a derelict space ship that inexplicably has a vintage fireplace that acts a 'time portal' into 1700s France, where he meets Reinette. The androids who own the ship had gone to 1700s France intending to use Reinette for some purpose related to repairing their ship.
Silence in the Library. A massive, abandoned library that takes up a whole planet is infested with carnivorous lifeforms that mimic shadows. And the library is kept operating by a young girl and a doctor.
Human Nature / The Family of Blood. The Doctor is hiding in Victorian era Europe and has made himself human to throw off an alien family that has been pursuing him. The plan is to eventually become a Time Lord again and retake his ship, but this becomes problematic when his new identity does not want to surrender his existence, especially with the life he has made for himself.
Utopia. After traveling into the far, far future when humanity is planning its last hope to prolong its existence in a 'dying' universe, The Doctor comes across an old enemy who shares his likeness.
Midnight. On a planet where the inhabitants must constantly be shielded from its brutal surface, The Doctor takes a shuttle ride with human passengers to visit one of the marvels of the planet, but when the shuttle is stopped mid-way toward its destination, it falls prey to a life-form that takes over its host and slowly learns to become more like it.