0-This really wouldn't affect Atlantis for a few seasons, but I would've had the Stargate go public after The Lost City, which would affect the type of aid received by the expedition after they inevitably reestablish contact with Earth.
1- Make Atlantis larger, about 21km in diameter, with facilities to totally rebuild a civilization in another galaxy. I would've also had be a lot more sturdy than the show has portrayed it and totally one of a kind, not just another city ship that the Ancients made.
2-The Wraith are biological intergalactic communications system gone bad, due to floating between galaxies for thousands, maybe millions of years. They ended up in Pegasus shortly after the Ancients seeded the galaxy with life and found the human population to be a convenient resource, useful for food and as the basis of the main computer of hiveships and cruisers. They wouldn't be introduced until the midseason two-parter.
3-The Atlantis expedition would have more international members in the military and civilian forces and have brought along a lot of supplies (including naquadah nukes and zats) and some manufacturing capabilities, as the expedition was seen as a long term colonization effort by its planners. They wouldn't regain contact with Earth until the end of season three.
4-Colonel Sumner would be British SAS and wouldn't die until the end of season one, which would be "The Storm."
5-Sheppard would be more like Yossarian from Catch 22, because of what happened to some of his friends in Afghanistan.
6-Ford would be some guy who's got more experience than Sheppard dealing with friendly offworld groups, but pretty green in terms of combat. McKAy would be the same.
7-The Athosians would be more of a nomadic group, staying on a world only long enough to harvest some crops before heading to one of the few safe uninhabited worlds to avoid the Wraith. Teyla became one of their leaders, along with Halling, after the old leaders died from some disease a few months before the expedition came to Pegasus. The Athosians would stay on Atlantis after the place they were living at gets hit with a flash flood, since it provides a significantly higher standard of living than 90% of the galaxy.
8-The introduction of the Genii would occur in the first part of the midseason two parter, as a pragmatic bunch of humans with the agrarian front and underground civilization. Sheppard's team and a bunch of Genii commandos go to the Keeper's hive and manage to recover a hard drive and use a naqauadah nuke to vaporize the ship, but miss the nearby cruisers, which send a general message to all Wraith in the galaxy. Knowing that it will take up to a few years for their message to be received (due to their slow subspace comm speed), the cruisers set out for Atlantis, the most likely threat.
9-The expedition decides to make allies with as many human socities as possible, including the people from "Poisoning the Well". Along the line, they visit a world under the control of the Confederation, a totalitarian regime that has mostly Genii level technology, but also has some Ancient technology.
10-Back on Lantea, the expedition finds an underground hangar with a specially fit Ancient destroyer (one of those smaller ships the Asurans use) for recon missions called the Hyperion. The onboard AI initially mistakes them for intruders and takes the ship into orbit, cutting them off from Atlantis until they convince the AI that they aren't hostile.
11-Koyla appears in "The Storm" as a Confederation spy in the Genii ranks, ordered to take Atlantis when it is evacuated because of the super hurricane. Koyla and a team of loyal soldiers pretend to be offering aid, then kill the soldiers guarding the Gate room. Colonel Sumner locks out the city's systems before being captured and tortured by Koyla.
12-Season 2 begins with "The Eye" happening pretty much as it does in the show.
13-Atlantis and the Genii become concerned about the threat that the Confederation poses and begin searching for its headquarters. Meanwhile, Atlantis continues building up its alliances in the galaxy.
14-Ronon appears in the first part of midseason finale: "Runner," as a Satedan soldier driven insane after spending seven years without a lot of human interaction, except for the Traveler who gave him his gun and a gate address that Ronon never went to because he feared that he would doom those people. After he gets the Wraith tracking device taken out of him, he asks to go back to Sateda to find out what happened to people placed in disaster shelters with supplies for ten years.
15- In part two, "Sateda," Sheppard and company take the Hyperion to Sateda and discover that the Satedans managed to damage the attacking Wraith cruiser enough to cause it to crash outside the capital. The remnants of the Satedan military have been fighting an uncoordinated guerilla war against the Wraith, which ends when the Hyperion blows up the cruiser and evacuates the Satedans to Atlantis, where Ronon is reunited with his wife, whom he married the day before the Wraith came.
16-The Satedans decide to remain on Atlantis, with the remnants of the Satedan military being lead by Ronon, who is the pretty much the only officer left in their chain of command, joining the expedition's miltary contingent.
17-The season 2 finale/season 3 premiere would have Sheppard's team getting captured by the Confederation and being sent to a gulag planet, where a group of Travelers are being kept after they contacted a man named Lucius, who happened to have an Ancient city ship. Ronon suggests they dial the address he got from the Traveler he met, which happens to be where Sheppard's team and the Travelers are. Ronon leads a Satedan/Expedition assault to free Sheppard's team and destroys the base after rescuing all the prisoners.