Early Stages of the Invasion
The only known early stages of the Secret Invasion are that Skrull agents were sent to replace their targets and infiltrate human organizations posing as them, waiting for the right moment to strike also attack.
Realization of the Invasion
As the Skrulls had infiltrated every group, including the Avengers, Nick Fury has discovered the invasion and left S.H.I.E.L.D. to form a team called the Secret Warriors on their mission to look out any Skrull agent posing as any hero or villain by any means necessary.
The Invasion Begins
Almost immediately, the Skrulls had begun their attacks on their targets, severely damaging Earth's planetary defenses and neutralizing potential resistance.
After the Super-Skrulls had taken out the Avengers, Captain America (Skrull) had made his announcement to the world to surrender to the Skrulls, but was interrupted by the arrival of the real Captain America, Thor, Iron Man, and the other heroes also the villains.
Then an all-out battle had broken out, when the heroes have gained the upper hand, Veranke had ordered Lyle Getz (Skrull) to activate the fail-safe: Sonic-disrupting satellites in order to teach the humans a lesson by making them lose their hearing. But the satellites were destroyed by Thor. All of the Skrulls were defeated one-by-one.
Aftermath of the Invasion
Despite that, Iron Man have given his Skrull detection gear to Sydren, which he used it to mass-produce them for S.H.I.E.L.D. and S.W.O.R.D. to hunt down every last Skrull left on Earth and lock them up in Prison 42 for their actions.
In the comics
The events of the Secret Invasion are based on the eponymous Marvel Comics event of 2008. The background was much the same: the invasion was motivated by a prophecy which foretold Earth to be the Skrulls' new homeworld after Galactus consumed their original home, Tarnax IV. Having been exiled by King Dorrek VII for spreading panic amongst the populace with the revelation of the prophecy, Veranke survived the destruction of Tarnax IV, which was indeed devoured by Galactus, along with a handful of Skrulls. From there, they began to plan their next move in fulfilling the prophecy, by replacing key human individuals with an advanced camouflage method, perfected through bioengineering: not only the Skrulls could now also copy powers, they also went through strenuous mental conditioning in order to act as sleeper agents, believing themselves to be the people whom they replaced until an activation phrase ("He loves you") brought back their memories. Some events from this point on differ from the cartoon's canon: Veranke herself spearheaded the invasion by disguising herself as Jessica Drew, a.k.a. Spider-Woman (who was not featured in the series, so the role was given to Mockingbird instead), who at the time acted as a double agent for S.H.I.E.L.D. and HYDRA at Nick Fury's behest, just before "Drew" was drafted into the New Avengers, thus giving Veranke multiple fronts to work on.
The hero community first became aware of the invasion when the New Avengers confronted the Hand in Japan and killed Elektra, who acted as their leader (as was done in the series to the false Viper leading HYDRA); upon her death, the corpse turned back into that of a Skrull. Since even Wolverine's sharp senses were unable to realize she was not the real Elektra, the super-heroic community, already fractured in the wake of the Civil War over the recently passed Superhuman Registration Act, became increasingly paranoid that anyone in their midst could be an infiltrator.
Along with Veranke (who, by then, as Jessica Drew, had accepted to work with Iron Man's pro-SHRA Mighty Avengers), the other main figures of the invasion were the Skrulls posing as Hank Pym (who was in control of the 50-State Initiative, a series of registered teams assigned to each State in the United States) and Dum Dum Dugan (who had stayed in S.H.I.E.L.D. even after Tony Stark was appointed as its director). A Captain America imposter was seen only briefly, as part of a team of sleeper agents posing as past incarnations of Earth's heroes which were sent to the Savage Land (much like the ones encountered by Ms. Marvel in Wakanda in the cartoon, their cover story was that they became stranded in space after trying to stop the Kree-Skrull War); he was confronted by Spider-Man, Ka-Zar and Shanna, who killed him.
When the Invasion was fully revealed, the heroes banded together for a final offensive against the Skrulls in the Central Park, in New York City. While Thor managed to destroy most of the Skrull fleet, the Wasp had been secretly infected with a mutagenic agent and, once the Skrull Hank Pym removed its failsafe, she started to grow out of control and release energy, whose pressure her body could not contain even at a great volume, i.e. she was turned into a living bomb. While she perished in the blast, Thor managed to contain the blast so that none of the other heroes would be harmed by the explosion. With their invasion plans crushed and their fleet decimated, the heroes converged on Veranke, who ultimately was murdered by Norman Osborn (a.k.a. the Green Goblin) with a weapon especially designed to kill Skrulls.