I'll echo those who recommend starting with Eccleston's run in 2005. I watched DW back in the 80s, but didn't start watching the new stuff until about 2 years ago; now I have my wife watching them, and it's a thrill that she's enjoying them.
I think all the episodes are worth watching at least once. (And picking just a few episodes out of each series to watch loses some of the emotional investment you develop for the characters, I suppose.) But if one person's opinion of the best episodes would be of help to you, then here goes. Some stories are split across two episodes.
SERIES 1 (2005): Eccelston/Piper
Ep 1: "Rose"
Ep 6: "Dalek"
Ep 8: "Father's Day"
Eps 9 & 10: "The Empty Child"/"The Doctor Dances"
Eps 12 & 13: "Bad Wolf"/"The Parting of the Ways"
SERIES 2 (2006): Tennant/Piper
Ep 3: "School Reunion"
Ep 4: "The Girl in the Fireplace"
Eps 5 & 6: "Rise of the Cyberman"/"The Age of Steel"
Eps 12 & 13: "Army of Ghosts"/"Doomsday"
CHRISTMAS 2006: Tennant/Tate: "The Runaway Bride"
SERIES 3 (2007): Tennant/Agyeman
Ep 1: "Smith and Jones"
Eps 8 & 9: "Human Nature"/"Family of Blood"
Ep 10: "Blink"
Eps 11 & 12 & 13: "Utopia"/"The Sound of Drums"/"Last of the Time Lords"
SERIES 4 (2008): Tennant/Tate
Ep 1: "Partners in Crime"
Ep 3: "Planet of the Ood"
Ep 7: "The Unicorn and the Wasp"
Eps 9 & 10: "Silence in the Library"/"Forest of the Dead"
Eps 12 & 13: "The Stolen Earth"/"Journey's End"
CHRISTMAS 2008: Tennant: "The Next Doctor"
CHRISTMAS 2009: Tennant: "The End of Time", Parts 1 & 2
I'm working my way through Series 5 right now, but I'm confident in recommending "The Eleventh Hour", which is Matt Smith's first adventure as The Doctor, and "The Lodger."
SERIES 6 (2011): Smith/Gillan
Eps 1 & 2: "The Impossible Astronaut"/"Day of the Moon"
Ep 4: "The Doctor's Wife"
Eps 7 & 8: "A Good Man Goes to War"/"Let's Kill Hitler"
Ep 12: "Closing Time"
Ep 13: "The Wedding of River Song"
CHRISTMAS 2011: Smith: "The Doctor, the Widow, and the Wardrobe"
If that list is too daunting, then go with these 5: "Rose", "The Girl in the Fireplace", "Blink", "Eleventh Hour", and "The Doctor, the Widow, and the Wardrobe."
All very much IMO, of course!