HL1: It's unclear whether G either caused or tried to prevent the resonance cascade event at Black Mesa, but it's clear that he knew what would happen (his warning to Eli that fateful day being "Prepare for unforseen consequences"). From that moment on, he appears to "evaluate" a number of people, seemingly disappearing and reappearing at will, saving some of them from certain death and vanishing moments later. It's apparent that these individuals hold some sort of promise that G appreciates, and wishes to see them progress against the seemingly impossible odds that the Xen-Earth conflict represent.
Through a combination of luck, bravery and sheer badassery, a single theoretical physicist named Gordon Freeman not only fought his way through swarms of hostile aliens and half a regiment of U.S. Marines out to kill him, but he managed to make it to the borderworld Xen, and destroy the master of the Xenite invasion, the Nihilanth. While this ostensibly saved the world from a massive invasion from Xen, it set the stage for far worse- the Combine. G, knowing that the excrement was seriously about to strike the ventilation device, sent Gordon into a "slow teleport", trapping him outside linear time for nearly two decades to ensure he could be placed exactly where "the right man in the wrong place" could be most effective.
Between HL1 and HL2: G takes notice of Eli's young daughter, Alyx. Even as a child, her future potential is clear to G, and through as-yet unknown means he rescues her from Black Mesa's destruction and works to ensure her survival of the Portal Storms, the Seventeen-Hour War and the Combine's takeover of Earth.
HL2: G re-introduces Gordon to the world he worked so hard to save, now under Combine dominion. As Gordon does his "tell me where to go and stay out of my line of fire" thing, G stays one step ahead of him, monitoring his progress and ensuring that he has the resources he'll need to proceed. He can be seen talking to Odessa Cubbage at New Little Odessa, possible briefing Cubbage on Gordon's approach. With the destruction of Nova Prospekt, the Resistance rises up, and with their help Gordon and Alyx storm the Citadel and stop a massive Combine incursion, destroying the Citadel's portal generator and apparently killing Dr. Breen in the process. As the generator starts to explode, causing a fireball that nothing could possibly survive, G "pauses" time and plucks Gordon away, sending him back into the slow-warp nullspace, speaking of "numerous offers" for Gordon's "services". Though he appears to leave Alyx to die on top of the Citadel, he makes it clear in Ep2 that he in fact values her even more than Gordon, and it is not beyond the realm of possibility that he rescued Alyx the same way as Gordon.
(A side note here- one of the lines that the Vortigaunts gives is "We serve the same mystery". It's possible that G holds some sort of sway over the Vorts through the events of HL2. However, this changes just ahead.)
HL2 Ep1: G steps back into the slow-warp nullspace to speak to Gordon again. This time, however, he is blocked by a number of Vortigaunts; at first this amuses him, but once he realizes that they are somehow preventing him from reaching Gordon he becomes angry for the first time we have seen. As the Vorts surround Gordon and return him (and presumably Alyx) to the foot of the Citadel, G speaks his only line in Ep1: "We'll see about THAT." It seems that G had plans to place Gordon elsewhere, but the Vortigaunts sought to return him to Earth to continue the fight against the Combine there.
HL2 Ep2: G appears to Gordon at the moment the Vortigaunts are struggling to save a severely injured Alyx's life. He reveals that they have been blocking him from contacting Gordon, but he managed to take advantage of their moment of distraction. He mocks their supposed dedication to humanity's cause, and reveals his monitoring of Alyx since her childhood, as well as Breen's knowledge of this (his distorted picture appears behind G during the visual sequence at a Black Mesa desk). He admits to having his hands somewhat tied by another unknown entity (certain "restrictions", as he calls them) and is reduced to simply asking Gordon to protect Alyx as they make their way to White Forest.
