So, where did this Crisis begin?
Flash of Two WorldsIn Flash #123 (September, 1961), Barry Allen met the Flash. Jay Garrick, that is, the original Flash, who premiered in Flash Comics #1 (and Barry had been reading his adventures in comic books for years, check in Barry's first appearance, and the time he became the Flash in Showcase #4).
This story by Gardner Fox and Carmine Infantino introduced the idea that the previous Flash (who had his own series for 104 issues) existed in a parallel world from the current comic Flash, and expanded the idea of what the DC Universe really was.
While helping Iris West at a fundraiser in Central City, Barry-Flash suddenly finds himself elsewhere....in Keystone City, the home of Jay-Flash. Barry goes in search of Jay, finding him (now slightly older than the comics Barry use to read, which were written by Gardner Fox, and married to Joan Williams, Jay's girlfriend in the comics Barry read). Barry convinces Jay to come out of retirement and the pair of Flashes face off against some of Jay's old foes (the Fiddler, the Shade and the Thinker - more on these villains coming in later articles on the Injustice Society of the World), defeating them, and then eventually returning Barry to his home world.
Barry's world had a lot going on at that time, the Atom (Ray Palmer) was premiering in Showcase #34, Superman was facing off against Brainiac with the help of Congorilla in Action Comics #280, Rip Hunter was continuing to travel in time with his fourth issue, the Sea Devils (a group of divers led by Dane Dorrance) were getting their first issue after a three-issue run in Showcase, and even Batman was busy fighting alien menaces like in Detective Comics #295.
Flash #123 was also reprinted on its own quite a few times...
...most notably in 80 Page Giant #9 in April, 1965 (along with a few other Flash tales), and in the Greatest Team-Up Stories Ever Told (presented in both hard cover and tradepaperback forms in 1990...the year of CBS' Flash TV show!).
Double Danger On Earth
Jay Garrick returned in Flash #129 (June, 1962), ending up on Barry's world, and fighting a few of his foes (Captain Cold and the Trickster) as Jay was there to get a meteorite to save his world from a deadly radiation.
Barry proved a good host to his new friend, helping Jay get the meteor (and Jay helped Barry defeat the two villains as well) in this story by Gardner Fox and Carmine Infantino.
The two Earths still had not been named at this point, but an important difference between the Flashes was introduced...Barry was a member of the Justice League of America, and Jay had been a member of the Justice Society of America (and a reference to their last published case of All-Star Comics #57 of February-March, 1951) was made getting the current comic reading world a possible glimpse into more events in Jay's world.
Back in Barry's world, Barry was teaming up with Green Lantern (Hal Jordan) in Green Lantern's own title (Green Lantern #13), as well as in Justice League of America #12 (facing Dr. Light, and also dealing with the JLA's newest member, Green Arrow), Atom was just moving up to his first issue and facing Jason Woodrue the Plant-Master, Batman was having troubles with the Joker in Batman #148, and Superman, well, he had his hands full with many cases (including the one in Action Comics #289).
Flash #129 was reprinted a few times....
...Flash #178 (April-May, 1968) contains the story (or most of it, two pages didn't make it), along with a Flash/Kid Flash story and the second Flash-Green Lantern individual team-up, and, Showcase Presents: The Flash #2, has this story (as well as Flash #123 and the next story), but in black and white.
Vengeance of The Immortal Villain
Flash #137 (June, 1963) starts with Barry seeing lights over six cities on his Earth, and, being a comic fan, puts together the fact that these lights correspond to the home cities of six members of the Justice Society of America, and thus travels to Jay's Earth and Keystone City.
In Keystone City, Jay-Flash finds a light there, and is captured in a stasis cube (which Barry helps free him from in this story by Gardner Fox and Carmine Infantino). The pair of Flashes then fought Vandal Savage (an immortal cave man who had been fighting the original Green Lantern and the Justice Society of America, who last appeared in All-Star Comics #37 of October, 1947, as member of the Injustice Society of the World, and that issue is shown owned by Barry in this issue no less), defeating him, and freeing the JSA's Atom (Al Pratt), Dr. Mid-Nite, Green Lantern (Alan Scott), Hawkman (Carter Hall), Johnny Thunder and Wonder Woman, all of whom (along with Jay) decided that maybe they need to get back into the hero business, reforming the Justice Society (which happens with Justice League of America #21 of August, 1963, which is the story which finally names Earth-1 as Barry's Earth, and Earth-2 as Jay's Earth in "Crisis on Earth-One"!).
By this point, Atom had joined the Justice League, and was meeting with the Thanagarian Hawkman (Katar Hol) in Atom #7, the Justice League was facing bigger menaces like Spaceman X in Justice League of America #20, the Faceless Creature returned to menace the United Nations in Strange Adventures #153, the group of misfits that were the Doom Patrol had their first case as a team with My Greatest Adventure #80, and Superman, Batman and Robin were facing the Band of Super-Villains in World's Finest Comics #134 (that were forgotten after this) showing that the Earth really needed more heroes!
Flash #137 was reprinted a few times as well...
...in Flash #213 (March, 1972), with a cover by Neal Adams, and in the Justice Society of America 100-Page Super-Spectacular (which also had a reprint of All-Star Comics #57!).
The Coming Crisis
These three issues of Flash set the stage for the first Justice League of America and Justice Society of America team-up in Justice League of America #21 (August, 1963), which became an annual event, and introduced many Earths (like the Crime Syndicate's evil Earth-3 and the Freedom Fighters' Nazi controlled Earth-X) and reintroduced many characters (like the Seven Soldiers of Victory and Sandman's junior partner, Sandy)...and all of that and more became threatened with the Crisis On Infinite Earths!
This all just gets you to the start of the Crisis, but it's a good start, and over time, check back to see more details filled in....or, explore on your own, with two collections that contain all 3 Barry-Jay team-ups, and fill in some more holes with the Flash of Two Worlds Hardcover of 2009, that has all three Flash stories above, as well as 3 later Flash team-ups, facing the Shade in one from Flash #151, facing Abra Kadabra (with help from Dr. Fate and Dr. Mid-Nite) in the second from Flash #170, and the Golden Man with the help of Kid Flash in the third from Flash #173...or in the Crisis on Multiple Earths the Team-Ups #1 of 2006, which also has the three stories from above, as well as Flash #151 with Barry and Jay vs. the Shade, the two Dr. Fate and Hourman team-up (including the introduction of the Psycho-Pirate from Showcase #56), the origin of the Guardians of the Universe and the first appearance of the "cosmic hand" and hints of the beginning of the universe with a team-up of Green Lanterns Hal Jordan and Alan Scott from Green Lantern #40, the first Starman/Black Canary team-up from Brave and the Bold #61 (with the return of the Mist) and a solo Hourman story (from Spectre #7). But, read a little of what's presented here, and you'll be ready for a Crisis!