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Some more thoughts on specific things to watch and when:

• One of the other answers includes a list by Neil Gaiman, which made me think about the episode he wrote in the most recent series, Nightmare in Silver. This is a Cybermen story, and for the most part the Cybermen in the modern series are not really connected to the original series Cybermen -- unlike most of the other enemies, they completely rebooted the Cybermen, gave them a new origin story, and so on. So while for the rest of the enemies (particularly the Daleks and the Master) I'd suggest going and watching some of the classic stories about them before seeing the new ones, I don't think this really applies so much to the Cybermen. Except: Gaiman's story is actually about the classic series Cybermen, not the modern ones -- it is more like the original stories stylistically, and includes reference to a couple of plot points from the original series that haven't previously been incorporated into the modern series (i.e. the future war between humans and cybermen that is in the history of most of the original series episodes featuring them, and a weakness they have that was exploited a few times in the original series but not mentioned previously in the modern episodes). So, while there's no need to catch up on the cybermen before then, at least prior to watching Nightmare in Silver catch up on the relevant classic episodes: Tomb of the Cybermen, Invasion, Revenge of the Cybermen and Earthshock.
• Revenge of the Cybermen is the last part of a continuous sequence of 4th Doctor stories which begins with The Ark in Space and also includes Genesis of the Daleks. While it isn't critical to watch these in order, I recommend doing so, as they're all good stories.
• Earthshock, also mentioned above, is a particularly significant story. You'll want to have seen some of the stories from the season before it and earlier in the same season before watching it. The E-space trilogy (which consists of Full Circle, State of Decay and Warrior's Gate) is a reasonable place to start, and the next three episodes that follow it (The Keeper of Traken, Logopolis and Castrovalva) which also form a sequence of stories by themselves are worth watching at this point too. Kinda is divisive. Some people like it, others hate it. Read reviews to see if you think it'll be worth watching.
• The Master first turns up in the third doctor era, and has a few great stories there (Terror of the Autons and The Daemons are particularly good), but his best stories are in the 4th and 5th Doctors' eras. The Deadly Assassin is one of the best stories in the show (and one of only a handful that don't feature a companion). The Five Doctors is a classic in its own right. I'll avoid mentioning others, because the show's producers often tried to keep it a surprise that the Master would be in a story, even to the length of using aliases for the actors who played him rather than putting their names directly in the credits. You can look them up if you feel you're missing anything, but I'll note I've recommended most of them here anyway.

I'm going to disagree with the majority here. I think it is worth watching at least some of the classic series early, in order to get an appreciation for where the show has come from. I'm not going to suggest starting with it, but I think jumping back to it between watching modern episodes could be an enlightening approach (which I wish now I had taken).

I'd also note that it's quite possible to dip in and out of most of the earlier parts of the show without regard to order. It might make sense, then, to watch the episodes in the order where the ideas they contain become relevant.

My suggestion as to viewing order, therefore, is something like this:

• Watch Rose, the first episode of the 2005 series. This will give you a taste of what the modern series is like.
• Watch Spearhead from Space, the first episode of the third Doctor's tenure. This will make an interesting comparison, because the enemy it deals with is the same enemy that turned up in Rose.
• Continue watching the 2005 series up to (but not including) Dalek. Before watching that, it's time for a history lesson.
• Watch original series stories: The Daleks, Dalek Invasion of Earth and Genesis of the Daleks.
• Watch the remainder of the 2005 series.

By now you should be hooked. You have an appreciation of both the modern and the original series. Go back and fill in some of the gaps. Many of the older episodes aren't worth watching, particularly the black and white ones: a lot of Hartnell's episodes are slow paced and frankly boring. Most of Troughton's are missing (but do watch at least the highest few rated of the ones that are available: Tomb of the Cybermen and Invasion are both brilliant, and War Games is really good if you're up for a long story... it took me 5 days to watch, but was well worth it). After that things start improving, but for me the show really picks up around the time Baker took over, and maybe that's where it's worth watching it all from.

When you're skipping around between the classic episodes, you'll want a good guide to the episodes with details about ratings, which recurring enemies are involved, and so on. I use this one. It's not as detailed as others, but it includes the rankings of each episode in a poll conducted by Doctor Who Magazine a few years ago, which makes it very useful.

I'm going to disagree with the majority here. I think it is worth watching at least some of the classic series early, in order to get an appreciation for where the show has come from. I'm not going to suggest starting with it, but I think jumping back to it between watching modern episodes could be an enlightening approach (which I wish now I had taken).

I'd also note that it's quite possible to dip in and out of most of the earlier parts of the show without regard to order. It might make sense, then, to watch the episodes in the order where the ideas they contain become relevant.

My suggestion as to viewing order, therefore, is something like this:

• Watch Rose, the first episode of the 2005 series. This will give you a taste of what the modern series is like.
• Watch Spearhead from Space, the first episode of the third Doctor's tenure. This will make an interesting comparison, because the enemy it deals with is the same enemy that turned up in Rose.
• Continue watching the 2005 series up to (but not including) Dalek. Before watching that, it's time for a history lesson.
• Watch original series stories: The Daleks, Dalek Invasion of Earth and Genesis of the Daleks.
• Watch the remainder of the 2005 series 1.

By now you should be hooked. You have an appreciation of both the modern and the original series. Go back and fill in some of the gaps. Try interleaving modern and old stories (although be aware that the ends of a lot of the classic stories feed directly into the next story in sequence, particularly starting with Baker's episodes, but they did do this sometimes before). Don't try to watch everything: many of the older episodes aren't worth watching, particularly the black and white ones: a lot of Hartnell's episodes are slow paced and frankly boring. Most of Troughton's are missing (but do watch at least the highest few rated of the ones that are available: Tomb of the Cybermen and Invasion are both brilliant, and War Games is really good if you're up for a long story... it took me 5 days to watch, but was well worth it). After that things start improving, but for me the show really picks up around the time Baker took over, and maybe that's where it's worth watching it all from. The companion(s) have a lot of influence over the tone, and IMO Sarah Jane Smith was one of the best. She was companion for Pertwee's last series and the first series-and-a-half of Baker. Make sure you've watched at least some of these before you get to School Reunion (from the modern series 2).

When you're skipping around between the classic episodes, you'll want a good guide to the episodes with details about ratings, which recurring enemies are involved, and so on. I use this one. It's not as detailed as others, but it includes the rankings of each episode in a poll conducted by Doctor Who Magazine a few years ago, which makes it very useful.