See my older post on the topic for some rationale for why certain episodes were chosen and placed in the order they were.
First, I recommend you start with these 15 episodes. The first 5 work together very well, and I'm extremely happy with their order. I'm watching numbers 6 through 15 to ensure they all work together. You still get introduced to the missing episode problem at the end of the list.
By the end of this you should have an idea of what you like. You'll also notice I've updated the notes to take better account of missing episodes.
After this you'll have 4 options of where to go next.
Based on what you like, choose one of these four, and watch them.
It's quite possible you'll like all eras of who, and if so, my recommendation is to move on to this list next.
This list contains nearly every Dalek story, but again presents the problem if missing episodes at the end.
After this you may want to hut down the remaining Dalek episodes, or you may have a desire to see different enemies. My recommendation here is to find a Doctor you most like, and simply begin there, watching all their episodes that you've not yet seen, in order, first to last.
THE CLASSIC ERA
This list is designed for those who find they prefer the classic era of Doctor Who.
You'll notice many episodes from the 4th Doctor. This is because the 4th Doctor dominated the classic era. You'll also notice a lack of episodes from the 1st and 2nd Doctors; this is as to avoid any problems with missing episodes. After this you may wish to pick one of these Doctors (or even go back to the 1st Doctor) and watch all episode in order, choosing if you wish to skip any you've seen as part of this intro.
THE MODERN ERA
This list is for those who prefer the modern era of Doctor Who.
It presents a good mix of Doctors, and gives you a good introduction to the modern series.
Following this you may wish to simply see all modern era episodes in order, or perhaps start with a specific Doctor, skipping the episode you've already seen, or, re-watching them if they were fun.
This brings us to our final option;
It's also quite possible that once you've seen these 15 (or even after you've only seen the top 5!) that you will have a good idea of what you want to see more of.