Series 3 is probably not a common choice for favorite arc, or favorite series, or favorite anything. Most people seem to point at "Blink" and "Human Nature"/"The Family of Blood" as two outstanding stories in the series, but then discount much of the rest as good enough and turn to other series for their season-long story arcs. However, Series 3 takes two different stories and weaves them together, developing them in each episode of the season.
The story arc that most people associate with Series 3 is that of Harold Saxon. He's first mentioned in "The Runaway Bride" as the Minister of Defense who gives the order to shoot down the Racnoss Queen's Webstar. In subsequent episodes, we see more mentions of him - campaign posters, news reports, casual mentions by characters - until "The Lazarus Experiment", when Francine tries to inform Martha that Mr. Saxon has named the Doctor as dangerous, and we now know that Mr. Saxon knows who the Doctor is and is in some way opposed to him. From there, we find out in "42" that Mr. Saxon is actively opposing the Doctor, to the point of subverting Francine against Martha. Then, of course, the final three episodes of the series gives us the big reveal, that Harold Saxon is actually the Master and has been preparing to take over the Earth with the Toclafane, hoping to also capture and disable the Doctor as part of the deal.
The other story is that of Martha and the Doctor. Martha meets the Doctor in "Smith and Jones" and immediately falls in love with him. (I will admit that this aspect of her is the one thing I hate about this season. If they had postponed her falling in love until "The Lazarus Experiment" and had her, as a reaction to her mother's reaction to him, realize that she was in love with him at that time, it would have sat better with a lot of the audience, as well as made that story a lot better. Of course, in DW time, it still would have only been a few days after she'd first met him, but in subjective viewer time, six episodes later would have felt better.) The Doctor, on the other hand, had just come out of his relationship with Rose and had realized that he can't allow himself to get close to a human (he realizes a little how this hurts Martha in "Gridlock") and that the relationship had made him ignore his responsibilities (this is demonstrated by how lost he thinks he is without Rose in "The Shakespeare Code"), so he overreacted and clammed up emotionally. Thus, Martha was seeking something he wouldn't give, and the Doctor became distant and callous. Without realizing it, he demanded more and more from Martha, until she finally stood up for herself and got out - possibly the best character growth in the entire show.
What's interesting is that even episodes that didn't seem to have much to contribute to either of the two story arcs were still essential to the series. "Blink" and "Human Nature"/"The Family of Blood" are both stories in which Martha must work hard and sacrifice for the Doctor but don't seem to otherwise contribute to the stories. However, "Human Nature"/"The Family of Blood" introduces the concepts of the chameleon arch and the perception filter, two important concepts in the final three episodes. If the audience had not been seeded with the idea that Time Lords could become human, the idea of the Master surviving the Time War by becoming human would have been ludicrous.
"Blink" is little more meta. In most stories, the Doctor tries to preserve the history that he already knows has happened. For example, in "The Shakespeare Code", he teaches Martha that if they don't stop the Carrionites, the history that she knows will never happen and she won't come to exist. "Blink" is different. In it, the Doctor goes back to 1969 and does things to set up the future that happens in 2008 so that all the right things happen to send the TARDIS back to him. This sets the audience up to understand the concept of the Master going back in time from the end of the universe to 2008 to set up the events that happen throughout Series 3. Most stories have reactive time travel, but "Blink" prepares the audience to understand proactive time travel.
The only story in Series 3 that I haven't mentioned so far is "Daleks in Manhattan"/"Evolution of the Daleks". It's really part of a different story arc, the story of the Cult of Skaro that is started in "Army of Ghosts"/"Doomsday" and ends in "The Stolen Earth"/"Journey's End". However, there are parts of it that contribute to Martha's story: Tallulah's love of and faith in Lazlo and ultimate reward could have influenced Martha to hold on to her hopes for the Doctor longer than she should have.
Thus, every single episode in this series contributed to the two story arcs, which spiral together into the season finale. To me, this was masterfully wrought. The frequent mentions of Mr. Saxon and the glimpses of his manipulation of Francine gave the audience enough information to speculate on what was going, while also watching Martha and the Doctor both degenerate until they force each other to grow up. On rewatch, you see all the clues and how they interweave. I much prefer it to the "here, have some random mentions of things, but you won't connect them until we tell you how to do so" style of the first ("Bad Wolf"), second ("Torchwood"), fourth ("bees disappearing" and "lost planets") and eighth (Missy) series, or the "here's a random glimpse of something the Doctor knows but can't find any clues about" style of the fifth (the crack), sixth (Kovarian's eyes and Amy's pregnant/not-pregnant state), and seventh (the Impossible Girl). Series 3 tells two separate but interrelated stories without relying on a bait-and-switch or distracting the audience with temporal trickery, and that's why it's my favorite.