Here's the super-short version:
Once upon a time, Earth peeps try to explore Mars and fail. But the pioneer spirit triumphs, and eventually they succeed in colonizing the planet. When war breaks out back on Earth, everyone goes back to be with their families. At the very end, some humans come back to Mars to get away from Earth's wars and start over.
Here's a slightly longer version:
"Rocket Summer": People on Earth make rockets. (But not in the summer.)
"Ylla": Telepathic Martian wife Ylla dreams about an Earth Man flying to Mars; Ylla's husband Yll is jealous; Ylla's husband kills Earth Man. Turns out that marriage is as tough on Mars as it is on Earth.
"The Summer Night": Some Martians randomly start playing popular American songs and acting like Americans. Evidently they're subconsciously picking up that a ship full of colonizers is on its way. Spooky.
"The Earth Men": Hang on, Shmoopers, because this one's a little complicated. The Earth Men finally show up, and the Martians are like, "Whatevs." Turns out, the Martians think that the Earth Men are actually just hallucinating Martians, since Martians can somehow project their hallucinations. So, the non-crazy Martians kill the Earth Men, because they think they're just crazy Martians.
"The Taxpayer": A man on Earth wants to board the next spaceship heading to Mars—and who can blame him, since parking is so expensive.
"The Third Expedition": And we're off to Mars again.The next ship of Earth Men finds an Earth town full of people they know who had already died back on Earth. Hooray! It's like Heaven on Mars. Except not, because actually the whole thing is a Martian trap.
"—And the Moon Be Still As Bright": The Fourth Expedition finds that most of the Martians have died from chickenpox. One guy on this expedition (Spender) thinks people shouldn't be there, so he kills a bunch of humans and they kill him back.
"The Settlers": The first group of people come to Mars to settle, kind of exactly like Europeans coming to settle the Americas.
"The Green Morning": The air on Mars is too thin for breathing, so one guy decides it's his job to plant trees to make oxygen. Boy, is he in for a big (and fast-growing) surprise.
"The Locusts": Even more people come to Mars, just like a swarm of—you guessed it—locusts.
"Night Meeting": A nice Earth Man named Tomás Gomez meets a nice Martian named Muhe Ca and they manage to talk without killing each other.
"The Shore": The second wave of Martian settlers is mostly city folks looking for space. Oh, and some religious folks ("The Shore").
"The Fire Balloons": Episcopal priests come to Mars to try to convert the Martians, who look like blue spheres. Turns out, the Martians are already beyond sins. Well, duh: what kind of trouble can blue spheres get into? This story is a 1997 edition exclusive.
"The Interim": Hey, look, it's like an entire Earth town got dropped in the middle of Mars.
"The Musicians": Boys will be boys, and in this case boys will play among the bodies of dead Martians.
"The Wilderness": Two women prepare for their trip to Mars, which is a lot like, you guessed it, the United States' Old West. Another 1997 exclusive.
"Way in the Middle of the Air": African-Americans in the South would rather go to Mars than deal with white racists. Pre-1997 editions only.
"The Naming of Names": Earth Men rename places on Mars.
"Usher II": A rich guy angry about censorship builds a haunted house that he uses to kill censors. (He's really angry.)
"The Old Ones": Now that Mars is safe, old people come.
"The Martian": A Martian shapeshifts into the long-dead son of an older couple. When they head into town, he can't help shapeshifting into the loved ones of everyone he passes. It's too stressful to be everything to everyone, and he dies.
"The Luggage Store": A luggage-store owner talks about how a war on Earth might make people go back—because obviously the first thing you do when you hear about impending nuclear war is head straight into the hot zone.
"The Off Season": A jerk from the Fourth Expedition opens a hot-dog stand and gets into a fight with some Martians who just want to talk to him. They end up giving him land on Mars, but the war on Earth means that he's not going to be able to profit. Serves him right, we say.
"The Watchers": People on Mars come out to watch the war and think about going home, which seems totally sensible. Not.
"The Silent Towns": After everyone else has left, the last man on Mars meets the last woman. It turns out that he really wouldn't marry her if she were the last woman on Earth—er, Mars—and so he escapes to live alone for the rest of his life.
"The Long Years": Another person who thinks he's the last man on Mars builds a robot family after his real family dies. Bummer.
"There Will Come Soft Rains": Back on Earth, a fully automated house goes through its last day before breaking down, thanks to just a teeny-tiny little nuclear Armageddon.
"The Million-Year Picnic": Back on Mars, one—or maybe two—families have escaped the nuclear war on Earth and come to settle on Mars. One has three sons; another has four daughters. Hm, wonder how the math is going to work out on this one?