If you're really wanting to get into Doctor Who, I'd recommend watching the "new" Series 1 (with Christopher Eccleston), then Series 2-4 (with Tennant), then the 4 Specials ("The Next Doctor"-"The End of Time," with Tennant), and finally Series 5 (with Matt Smith.) Then think about checking out some classic Who, which is, IMO, more difficult to get into.
If you don't want to dive in headfirst like that, here are the episodes I'd recommend you check out to see if it's your cup of tea (all from the 2005-present revival) - the bolded episodes are the particularly good ones:
Series 1
Rose
The End of the World
The Unquiet Dead
Dalek
Father's Day
*The Empty Child / The Doctor Dances (two-parter, my personal favorite episode)
Bad Wolf / The Parting of the Ways (two-parter, season finale)
Series 2
The Christmas Invasion (David Tennant's first episode; I'd also recommend watching the 7-minute Children in Need special before this)
New Earth
Tooth and Claw
*The Girl in the Fireplace
Rise of the Cybermen / The Age of Steel (two-parter)
The Impossible Planet / The Satan Pit (two-parter)
Army of Ghosts / Doomsday (two-parter, season finale)
Series 3
Smith and Jones
Gridlock
Human Nature / The Family of Blood (two-parter)
*Blink
Utopia / The Sound of Drums / Last of the Time Lords (three-parter, season finale)
Series 4
Partners in Crime
Planet of the Ood
The Sontaran Stratagem / The Poison Sky (two-parter)
*Silence in the Library / Forest of the Dead (two-parter)
*Midnight
Turn Left / The Stolen Earth / Journey's End (three-parter; parts 2 and 3 are crossover episodes with Torchwood and The Sarah Jane Adventures; season finale)
Specials
The Next Doctor
The Waters of Mars
The End of Time (two-parter, Tennant's last episode)
Series 5
*The Eleventh Hour (Matt Smith's first episode)
The Beast Below
The Time of Angels / Flesh and Stone (two-parter)
*Amy's Choice
The Hungry Earth / Cold Blood (two-parter)
Vincent and the Doctor
The Pandorica Opens / The Big Bang (two-parter, season finale)
* = episodes that have, in my experience, "hooked" people on the show