While I think Rose is only a middling companion, I really enjoy Christopher Eccleston's year as the Doctor. For a show that was pretty much dead for 16 years, that season really brought it roaring back, with a variety of time periods & interesting stories. They pulled out all the stops here, particularly since there was no guarantee that the show would continue from here. It's an excellent introduction to one of the most unique sci-fi shows in the history of TV.
It's important to remember that Doctor Who cannot be easily categorized into a sci-fi sub-genre because there's never been anything else like it. It's not a spaceship/military show like Star Trek, Stargate SG-1, or Babylon 5. It's not a paranormal investigation show like The X-Files or Fringe (although the Doctor Who spin-off Torchwood is kinda like that). It's not an anthology series like The Twilight Zone or The Outer Limits. It's a different beast altogether. Yes, the Doctor explores strange new worlds, but he does so on his own, without any sort of official backing or even much of a crew. (He often travels with one or two human companions but they rarely offer anything in the way of alien expertise.) The Doctor also travels without any sort of specific mission. He's an alien with wanderlust who mostly just tries to do the right thing.
I would warn most modern audiences to stick with the modern series, then explore the older series as your interest grows. The old series can be a very acquired taste in terms of pacing (slow) and production values (cardboard). (Seriously, I cannot warn you enough. If you are a special effects snob, you will probably hate the old series. It's so chintzy, it makes the original Star Trek look REALLY GOOD by comparison.) However, if you can acclimate to the different style of storytelling on the original 1963-1989 series, there are many fine stories to be had and even better acting to watch. Patrick Troughton & Tom Baker are still in a dead heat for my all-time favorite Doctor (although the new boy, Matt Smith, is making up ground very fast).
The new series is very arc based. You would probably be best served beginning at the start of Christopher Eccleston's 2005 season with "Rose" and go forward from there. If you don't want to start so far behind, Matt Smith's 1st episode "The Eleventh Hour" is a decent introduction.
OTOH, the older series is largely made up of a bunch of stand-alone serials. You can skip around that era willy nilly and not really miss anything. There were only a few more involved storylines that the old series did:
The UNIT years (Seasons 7-10): During most of Jon Pertwee's run as the Doctor, the TARDIS was crippled and the Time Lords forcibly exiled the Doctor on Earth. Instead of traveling through time & space, the Doctor works as a scientific advisor for an international military organization that specializes in alien phenomena: U.N.I.T., the United Nations Intelligence Taskforce. (In later years, the UN requested that they take their name off of it, so it became the UNified Intelligence Taskforce.)
The Key to Time (Season 16): During this season, the 4th Doctor (Tom Baker) is assigned by the Time Lords to seek out the 6 segments of a powerful alien artifact called the Key to Time. For the most part, this is just a McGuffin that leads the Doctor to various alien planets where he has largely unrelated adventures. The actual Key to Time only really achieves relevance in the final story when it is nearly completed in "The Armageddon Factor."
"The Trial of a Time Lord" (Season 23): During Colin Baker's final year, the Doctor is summoned by the Time Lords and put on trial for "violating the laws of time" or something like that. As evidence in the trial, the prosecution & the defense both provide video of the Doctor's adventures in different circumstances. (Generally, you can only buy the DVDs of this one in a complete box set.)