It depends on who you ask, but general historical consensus is that Tombstone's version, which has Billy Clanton and Wyatt Earp firing simultaneously, is the most accurate.
The Earp version is much as depicted in the film: the Earps and Holliday went to the OK Corral with the intention of disarming and possibly arresting the cowboys, after a series of confrontations between the two groups and further threats by Ike. Virgil claimed that due to Sheriff Behan's claims that he'd disarmed the gang, they were not expecting a fight, and were surprised when Frank McLaury and Billy Clanton were visibly armed (and they believed, though it has never been proved, that Tom McLaury was carrying a concealed weapon). Virgil tried to calm the situation down, but Billy went for his gun, initiating the gunfight. Wyatt and Billy fired simultaneously, according to this version; Billy fired at Wyatt and missed, but Wyatt fired at Frank McLaury and hit him in the stomach. After a few seconds, the firing became general.
The Cowboy/Behan version is that the Earps went to the scene spoiling for a fight, with Morgan reportedly shouting "You sons of bitches want a fight, well now you've got it!" or words to that effect. According to this version, Holliday fired first, either with his shotgun, or with a "nickel-plated" pistol, and one of the Earps, probably Morgan, fired shortly thereafter. In this scenario, the Clantons were effectively murdered by the Earps, trying to surrender as the Earps opened fire, and only returning fire in self-defense. Indeed, Ike Clanton, along with Behan and relatives of the McLaurys, would attempt to bring murder charges against the Earps, but these were dismissed in a pre-trial hearing, setting the stage for future confrontations between the two groups.
Eyewitnesses gave conflicting accounts of exactly how the shooting started, but the weight of evidence is in favor of the Earp version. For one thing, the sickly, thin-framed Holliday would probably not have been able to fire a pistol while also cradling a shotgun, and the claim of his firing a concealed pistol to start the shootout is unlikely. That Ike Clanton survived the shooting unscathed also cast doubt on the Cowboy depiction, though Ike would claim he physically attempted to prevent Wyatt from firing. Conflicting testimony by pro-Cowboy witnesses, especially Ike, during the hearing would also help exonnerate the Earps.