Look for the mission called Priority: Eden Prime. Assuming you have the DLC installed, this should be available to you in the game as soon as you board the Normandy and can use your Galaxy Map.
It is, in fact, the first thing I did upon having a choice about where to go.
As a heads up, there is a bonus mission on that planet called Eden Prime: Resistance Movement. Make sure you've done everything possible on that before completing the main DLC mission, as it appears you can not return to Eden Prime later if you missed anything.
The Illusive Man is the leader of Cerberus. He is a shadowy figure that few people meet in person. In Mass Effect 2, he "recruited" Shepard and gave him/her the tools needed to investigate the Collectors.
For instance, the version of the Normandy you control in ME3 is Cerberus-built at the Illusive Man's direction, and Shepard has certain implants that were manufactured by Cerberus as well. Some of the people you'll meet in ME3 are active/former employees of the Illusive Man. Miranda in particular was a key Cerberus operative/Illusive Man informant during the events of ME2.
However, he and Shepard parted ways at the end of ME2. Depending on your actions in that game, he can be more or less angry by the conclusion of ME2, but it seems that no matter what you do, the Illusive Man and Cerberus will be your adversary in ME3.
You could classify him as an "evil genius" - he's got good ideas and wants to ensure the survival of humanity, although his methods are ethically questionable. He doesn't trust other races, and is actively hostile towards them at times. As Shepard's goal is to unify the Council races against the Reapers, he/she is frequently at odds with the Illusive Man's methodology and tactics.
His tendencies to encourage brutal experimentation on living creatures, his desire to control, engineer and evolve humanity into the dominant race in the galaxy, and his rampant xenophobia draw strong parallels to other historical tyrants. It's arguable that he represents as real a threat to the galaxy as the Reapers do.
Typically he is seen sitting in a chair in a large room, surrounded by holographic displays. He frequently smokes, and has bright blue eyes. His appearance is modeled after Martin Sheen, who provides his voice. His backstory is left mysterious in the games, but has been expanded upon in other literature. The comic series Mass Effect: Evolution contains a good portion of his backstory.
Also, don't confuse "Illusive" with "Elusive" - it's arguable he's both, but I think many people think of the latter term when they think of his name.
Best Answer
I'd say yes. The story evolves over the course of the previous two games, and there's really no way to watch/play even the introduction sequence of this game without spoiling some of the major plot reveals of the first.
Things that Shepard discovers in the first and second games are pretty much widely known at the beginning of the third, so the plot twists and turns of those games will be completely lost as people are going to be conversing about them at length even early on.
Some of the returning characters played major roles or were scarred by events in the previous game ("the Virmire survivor" comes to mind as being key), so attempting to talk to them without getting spoiled about the events of previous games is virtually impossible.
That's not to say that starting here will utterly ruin your taste for either of the previous games. I believe Bioware has tried to make this chapter accessible for newcomers, and the plot of the first two is nuanced and complex enough that you'd still gain from going back and playing the first two if you get engrossed in this game's universe.
However, if you're the type that values not being spoiled, starting with ME3 is going to really reveal a ton about the plot of the first two games.