Man, daunting is an understatement. 😉. I’m inclined to agree with oojason on this one. Start with Rose, 2005 and see how you like it.
That said, as it went along and regained its footing after the decade + off, it got better. If I were to pick random episodes, I’d go with these (below). When I get to work Monday I can cut and paste a detailed list with explanations I made for a coworker wanting to jump in the way you are wanting.
Blink. A perfect first episode because it’s told from the point of view of the main character, Sally Sparrow. As a new viewer, you’re at as big a loss of understanding as she is. In fact, the Doctor is hardly in the episode. Bonus: Carey Mulligan. swoon
Vincent And The Doctor. Like Blink, a great stand-alone episode where the story is more important than time travel. One of my favorites.
The Eleventh Hour. A wonderful first episode because it’s the 11th Doctor’s first episode. A great look at how he struggles after the regeneration to adjust to his new body and self. I’d put the first 20 minutes or so - The Doctor and the little girl - up against anything in Hollywood for heartwarming. When it’s just The Doctor and her meeting, cooking, and sitting at the table eating - perfect.
Journey To The Centre Of The TARDIS. Another good stand-alone. This is probably the episode where I fell in love with The Impossible Girl.
There are others that I like as much or more, but may not be great first episodes because they’re a little heavy.