It depends quite a bit on the precise terminology you mean.

Suicidal attacks, in which an individual or group launch some kind of attack in which their death is all but guaranteed, are ancient and in all likelihood pre-historic. For instance, given the odds involved, the 300 Spartans had to have been reasonably certain that they were not going to return from Thermopylae alive. The odds were "suicidal".

But a "suicidal" attack, in which ones death is an inexorable but still entirely incidental feature of the tactic being employed, is a distinct topic from a "suicide" attack, in which the very tactic being employed inherently involves the death of the individual/s involved, or, in even more stark examples, where the death of that individual is actually a desired thing on the part of the attack. So for example we might characterize instances where soldiers are ordered to fight to death as a "suicide mission," but if they happen to survive, bene. Whereas, say, for a Japanese Kamikaze pilot, the act of the pilot dying is an intrinsic feature of that tactic being viable. Or, in the starker version that I mentioned, you could look at Islamic suicide bombers where not only is death intrinsic to the attack, but the idea of martyring oneself as a guaranteed route to heaven is actually regarded as a positive feature of the attack to be celebrated.

Now, with kind of the terminology of this out of the way, in the example that you gave:

(a guy walks into an establishment, puts a bomb down, starts it going, and then I guess had no time to leave before it would go off).

Now, this specific scenario sounds to me like it's either A: accidental, or B: a suicidal attack, but it does not sound to me like a suicide attack.

Now, if it is accidental, which from your description seems most plausible, those kind of incidents are common very early indeed. The Shakespearean expression "hoist with his own petard" refers to a sapper accidentally blowing himself up with his own bomb. So already in about 1600 not only is this a known concept, it's one that's familiar enough that it's being used a pithy metaphor by popular playwrights.

Unfortunately I can't point to the earliest example of a suicidal bomb attack. My suspicion would be that it was virtually contemporary with the invention of bombs. Suicidal missions strike me as simply being so intrinsic to warfare that I imagine that generals were ordering grenadier units to rush up to certain death and throw their grenades virtually from the moment the grenade was invented. I don't know that for sure, but it strikes me as being plausible given how ubiquitous that kind of attack is.

Now suicide bombing, as you asked about as your question is a separate matter. I once sat in the terrorism section of a pretty well-stocked academic book store and tried to answer this question for myself: "What was the earliest example we know involving someone strapping explosives to themselves or otherwise carrying out a bomb attack in which their death was intrinsic to the attack itself?"

After combing through books like, if I recall correctly, The Routledge History of Terrorism and The Encyclopedia of Terrorism, the best answer I was able to find of the earliest example of suicide bombing as defined above as a strategy is the Russian anarchists and nihilists of the late 19th and early 20th century.

In his book Blood and Rage: A Cultural History of Terrorism. Michael Burleigh writes:

In a further shocking development, the new-wave terrorists resorted to suicide bombings, in addition to attacks that were already a subliminal form of killing oneself. In 1904 terrorists connected to anarchist groups walked into gendarme or secret police buildings and blew themselves up. On 12 August 1906, three terrorists dressed as gendarmes tried to enter prime minister Stolypin's villa on an island near St Petersburg. The minister's guards held them in an antechamber, where, shouting 'Long live freedom, long live anarchy!', they blew themselves up with sixteen pound bombs.... In 1908 nine members of a terrorist group were arrested for plotting a suicide attack on the justice minister. One of their number wsa kitted out as a human bomb, the idea being that he would hurl himself beneath the minister's carriage, simultaneously detonating the bomb. When the police tried to arrest this Conradian figure, he warned: 'Be careful. I am wrapped around with dynamite. If I blow up, the entire street will be destroyed.' Seven of this group were sentenced to death and hanged."

This is perhaps secondary to your question, but since then suicide bomb attacks today are perhaps most commonly associated with radical Islamist groups. Islam has something of an unusual history in distinguishing a "suicide" attack from a "suicidal" attack because of the doctrines surrounding the Islamic concept of martyrdom in the context of jihad. The medieval debates on this are fascinating, and there's an incredibly contrast between the rewards given to the martyr in the afterlife (most famously, if perhaps numerically erroneously, sexual rewards with the 72 virgins of paradise), and the much more prosaic discussions of what legally constitutes martyrdom. One of the crucial such distinctions that's established the medieval period comes from Ghazali and is cited later by others that:

if one commits this action [i.e. seeking martyrdom] with the intention of creating terror (nikaya) in the herats of the enemy, then it is acceptable.... Other scholars tended to agree with al-Ghazali, and the general consensus was that if such actions were beneficial to Muslims in general, then they can be undertaken.

(from Martyrdom in Islam by David Cook.)

Nonethless, by the middle 20th century these ideas were not yet strong enough to prompt suicide bombings. Members of the Muslim Brotherhood in the 1954 considered mounting a suicide bomb attack against Nasser, but rejected it as being too extreme. They instead tried to shoot him during a speech, which failed, and backfired even further when Nasser's calm response cemented his reputation as someone willing to give his life for Egypt, propelling him to new heights of popularity. The attacker and five co-plotters would later be hanged.

To my knowledge the first systematic Islamist suicide bombings were carried out by the predecessor organization to Hezbollah during the Lebanese Civil War. At the time, the group potentially as something of a fictitious cover, called itself the Islamic Jihad organization. Inspired by the rhetoric around martyrdom in the context of jihad espoused by the Ayatollah Khomeini (and borderline, individual cases like Muhammad Fahmideh from the Iran-Iraq war), Hezbollah pioneered the use of suicide bombings ideologically, tactically (for instance, by launching attacks with a mix of chemicals in large, sophisticated truck bombings), and strategically. Their dual bombing of the US embassy and Marine barracks in Beirut not only decimated the CIA presence in Lebanon, it ultimately prompted US withdrawal from the country-- ultimately prompting other jihadists like Osama bin Laden to conclude that the US would concede to the political demands of such attacks swiftly given sufficient loss of life.

The first Sunni group to make widespread use of suicide bomb attacks was Hamas, starting in 1994. Expanding to be favored tactic of Palestinian terrorists generally, the tactic has since become virtually the signature feature of Islamist terrorist and militant groups.

There have been other tactical and strategic elements to suicide bombings contributed by other groups. Perhaps the most twisted is the "suicide bombing by proxy" which the IRA was noted for using.

Source wise:

well, aside from the sources I mentioned in the body of the post above, the anecdote about the attempted assassination of Nasser comes from Eugen Rogan's The Arabs: A History.