The 1868 Declaration Renouncing the Use, in Time of War, of Explosive Projectiles Under 400 Grammes Weight (also known as the Saint Petersburg Declaration of 1868) was really the first modern international treaty limiting a weapon for its gruesomeness.

The weapon banned were rifle bullets that exploded upon impact. The rationale was a big milestone in international law: when a soldier is hit by a bullet, he is no longer a combatant. Since the normal bullet by itself achieved its purpose, an exploding bullet causes unnecessary harm. This same language was used in later treaties.

It was initiated by the Russian Tsar Alexander II. His army had invented such a bullet, but knew that its secrecy wouldn't last long. So, he invited most of the European powers to St. Petersburg for the conference and they signed the treaty. No one has since used such bullets.

The Geneva Conventions (especially the update one in 1949) ban several different types of weapons and strategies.

These include any weapon or strategy which does not distinguish between civilian or military targets. Under that criteria, aerial bombing, nuclear weapons, and landmines are illegal.

Banning something, and enforcing the ban are different things though.

Have a read

The Laws of War, as we know them, are a relatively modern development. There have been prohibitions in the past, the most famous is the supposed banning of Crossbows against Christians by the Church, but whether that ban is authentic (and whether, if it was, it was followed) are a matter of debate.

I did a post previously about the development of the laws of war, which might help you.

OK. The modern conception of "Laws of War" really trace back to the 1600s and Hugo Grotius. His work 'On the Law of War and Peace' was one of the first of its kind, and essentially established the modern concept of laws of war, addressing both the idea of jus ad bellum - what are the reasons that justify going to war, and jus in bello - what is the expected conduct of soldiers when fighting.

That isn't to say that he was the first to conceptualize of these things, but he wrote them down in a succinct way that was influential well beyond his time. We can go back and see all kinds of writings that touch on the issue though, even the Old Testament talks about conduct in war, but someone else will need to talk about how they did it in ancient times.

So anyways, following Grotius, we start to see the evolution especially of jus in bello. Treatment of POWs and the enemy wounded are some of the big ones early on. The 19th century is where we start to see international agreements that really lay out explicitly what is allowed though. In 1863, the Lieber Code is issued by Abe Lincoln, dictating the expected conduct of Union Forces, and explicitly lays down many of the rules that we will later see in the Geneva and Hague Conventions, including treatment of prisoners, and who could and who couldn't be executed (Soldiers no, Spies yes). The Lieber Code wasn't really that important in the Civil War, but it was exceptionally influential in the writing out of later works.

The First Geneva Convention was agreed to a year after the Lieber Code, and and I believe was the first international agreement of its kind, signed by most of the major European powers. It was concerned with, as its title alludes to, "Amelioration of the Condition of the Wounded in Armies in the Field." A few years later, 1868, we see the St. Petersburg Declaration, which directly addressed acceptable ammunition, prohibiting small caliber (under 400 grams) ammunition that was designed to explode. Again, pretty revolutionary.

From there we see the Hague conventions in 1899 and 1906, which are the foundational treaties for our current laws of war. Although it had kind of banned the use of asphyxiating gas, this was obviously not followed in World War I, and the Geneva Protocol was added in 1925 to more clearly prohibit it in all forms. In 1929 the 'Geneva Convention on relative to the Treatment of Prisoners of War' comes about, offering better protections to them. The Soviet Union hadn't signed this though, and Japan hadn't ratified it, which led to it essentially being ignored by those nations during WWII - and Germany in regards to the Soviets.

We also can't forget the poor Kellog-Briand Pact, which attempted to ban war. We can be charitable and say it was a little ahead of its time (not totally untrue, as aspects of the UN charter echo it).

Following World War II, the Geneva Conventions underwent a major overhaul in 1949, which is what people generally mean when they say "The Geneva Convention". Three of them were updates on earlier agreements - that on sick and wounded, a later one that addressed sick and wounded at sea, and the POW one. The fourth convention addressed civilian populations, who obviously had suffered greatly in WWII. Since then, three protocols have been added, mostly dealing with victims rights, but also use of the Red Cross Emblem.

Since 1949, there have been a number of other agreements and treaties... dealing with nukes, chemical weapon stockpiles, landmines... you name it. I don't know them all off the top of my head though. But really, since 1949, the changes have been rather small all things considered. Geneva Conventions are still the gold standard of the Laws of War, so to speak.

The biggest change since 1949 I would say, is the perspective as to how the laws apply. In most cases, a nation can't break international law if they are not party to the treaty that establishes it, or if they registered an objection to a certain part. While all states (I believe) have signed and ratified the Geneva Protocols by now, not all have done so with the Protocols, and in theory, a state could repudiate the treaty. And of course there are other agreements that almost all nations have ratified, but a few refuse to. In international law, the idea of jus cogens, or of customary international law, is the concept that regardless of a treaty existing or being signed or ratified, there are simply some things that are never allowed, ever. In the post-WWII era, the idea that at least some laws of war fall into this category has certainly become more prevalent.

Now, as to "Which was the first war fought according to the "rules"?" That is a hard question to answer. Following the Hague Convention of 1899 coming into force in 1900, it would seem the first war fought between parties to the agreement would be Japan and Russia during the Russo-Japanese War.

Between parties to the Geneva Convention of 1864... It looks like the Austro-Prussian War would be it. Prussia ratified in in 1864, and Austria did so on 7/21/1866, a month into the war (might not be a coincidence...)

But that's the answer for modern laws of war, so there are certainly answers that are earlier.