I've actually discussed this question before! :D I'll go ahead and steal some of the stuff I wrote down there and put it here, while adding in a bit more context for you.

Funniest thing about elephants in antiquity - they were seen as the shock troop by those who used them (Lookin' at you, Carthage and the East). They were like monsters out of legend, and they were terrifying when charging. However, elephants were similar to a hand grenade without a pin - you never knew what would set them off, or even if they would hurt you more than they hurt your enemies. Again - Carthage is one of the greatest examples of an over-reliance on elephant warfare...and when facing off against the Romans, they never were able to use them to their full potential. Hannibal Barca is famous for bringing his elephants across the treacherous passes of the Alps - and yet, the only time they were extraordinarily useful was in the Alps, when the tribesmen, confronted with this monstrous creature, were scared off by them. Without that boon, Hannibal might not have made it.

However, we're discussing methods that the Romans used against elephants! So here goes nothing :D The first time that the Romans encountered war elephants was when fighting Pyrrhus of Epirus in Magna Graecia (Southern Italy). Pyrrhus is famed for his "Pyrrhic Victories" against the Romans (He won his first few battles, but with extremely heavy losses), and the first of these was accomplished with elephants. From Plutarch!

At last, when the Romans were more than ever crowded back by the elephants, and their horses, before they got near the animals, were terrified and ran away with their riders, Pyrrhus brought his Thessalian cavalry upon them while they were in confusion and routed them with great slaughter.

Needless to say, the Romans were perturbed by this loss. They hadn't ever faced elephants before, and those things are TERRIFYING if they're charging you down. So they had to come up with a solution. And they were a bit freaked out by how many men they'd lost - in fact, there were a whole bunch of Romans who advocated quitting the war! But then, an ancient man named Appius Claudius, who was EXTREMELY distinguished in his day, heard that Rome was about to concede. And he got PISSED. He had himself carried to the Senate (he was blind), and gave a speech, chastising the Senate as a whole, telling them that Pyrrhus was merely "The servant of one of Alexander's bodyguards," and that they were pathetic for cowering before him.

Needless to say, Rome changed their mind about the whole peace thing. They sent back the message that, so long as Pyrrhus "was there in arms, they would fight him with all their might, even though he should rout in battle ten thousand men like Laevinus. [the consul]"

The Romans then sent an emissary to discuss the matter of the Roman prisoners that had been taken, and it was headed by a man named Caius Fabricius, who was really poor. So Pyrrhus, trying to get a good deal out of him (and probably trying to get an in with regards to Roman politics) offered him a bunch of gold, which he rejected immediately. So then....

[...] on the following day, however, wishing to frighten a man who had not yet seen an elephant, he ordered the largest of these animals to be stationed behind a hanging in front of which they stood conversing together. This was done; and at a given signal the hanging was drawn aside, and the animal raised his trunk, held it over the head of Fabricius, and emitted a harsh and frightful cry. But Fabricius calmly turned and said with a smile to Pyrrhus: "Your gold made no impression on me yesterday, neither does your beast to‑day."

I had to tell that story. But anyways! So Pyrrhus had to fight another battle at Asculum against the Romans. The Romans had learned from their previous encounter that the elephants needed to be dealt with. Pyrrhus had apparently lost one, so he was at 19 of them - but that's still 19 elephants! And Rome was TOTALLY ready this time. Dionysius of Hallicarnassus gives a WONDERFUL description of them!

Dealing with Elephants: Attempt #1

Outside the line they stationed the light-armed troops and the waggons, three hundred in number, which they had got ready for the battle against the elephants. These waggons had upright beams on which were mounted movable traverse poles that could be swung round as quick as thought in any direction one might wish, and on the ends of the poles there were either tridents or swordlike spikes or scythes all of iron; or again they had cranes that hurled down heavy grappling-irons. Many of the poles had attached to them and projecting in front of the waggons fire-bearing grapnels wrapped in tow that had been liberally daubed with pitch, which men standing on the waggons were to set afire as soon as they came near the elephants and then rain blows with them upon the trunks and faces of the beasts. Furthermore, standing on the waggons, which were four-wheeled, were many also of the light-armed troops — bowmen, hurlers of stones and slingers who threw iron caltrops; and on the ground beside the waggons there were still more men.

So, how did these wondrous ancient tank/APC/battlewagon things fare against the elephant?

When the king had ordered the elephants seem to be led up to the part of the line that was in difficulties, the Romans mounted on the pole-bearing waggons, upon learning of the approach of the beasts, drove to meet them. At first they checked the onrush of the beasts, smiting them with their engines and turning the fire-bearing grapnels into their eyes. Then, when the men stationed in their towers no longer drove the beasts forward, but hurled their spears down from above, and the light-armed troops cut through the wattled screens surrounding the waggons and hamstrung the oxen, the men at the machines, leaping down from their cars, fled for refuge to the nearest infantry and caused great confusion among them.

So, those worked well. Ish. Other than the fact that the Romans needed men with the wagons to fight the men with the elephants. Supporting the elephants? What a NOVEL idea!

Dealing with Elephants: Attempt #2

Later on in the Pyrrhic Wars, the Romans learned a few things:

• Elephants don't like fire

• Elephants don't like pain

• Elephants don't like noise

So Roman strategies from here on generally included two of the above. Next time they fought Pyrrhus' elephants was at Beneventum - where the Romans defeated the Epirotes by shooting their elephants full of flaming arrows, causing them to panic, stampeding through the ranks of the Epirotes.

Dealing with Elephants: Part 3

The most famous incident of anti-elephant warfare was in the showdown between Hannibal Barca and Publius Cornelius Scipio (Later Africanus). Hannibal had vast numbers of decently strong, but inexperienced troops - and his best cavalry (the Numidians) had gone over to the Romans. He also had 80 elephants. So! Starting off, the Romans began with noise - the cavalry blew their warhorns all at the same time, causing a good number of the elephants to panic, stampeding through the Carthaginian lines. Next thing Scipio did was something that was very difficult for Romans of this time to do. Let me add some context here.

The manipular army of the Punic Wars was probably the greatest "levy/citizen/militia" army in history. It was more flexible than the phalanx, and way more useful than most of the other formations in use during this period. However, it wasn't perfect - mostly because of that whole "militia" aspect. Men were untrained and unused to working together, for the most part - and that meant that the Romans were limited in tactical flexibility with what their men could respond to.

Enter Scipio. He'd just come off of a strong campaign in Spain, and he'd had years in command of these men, who he'd trained hard, forging them into a unit, rather than just a consular army. THen, for his invasion of Africa, he pulled the same thing - except he added some invaluable men to his army. He added the survivors of Cannae, veterans who had seen the worst come to life. And they also had a vendetta against Hannibal - they'd been exiled for losing to him.

So, all of these well-trained veterans were super well-coordinated - most especially the most notoriously flighty Roman troops, the velites, or the light skirmisher infantry. So what Scipio did at Zama was he had his heavy infantry form channels - and filled the channels with these velites. When the elephants charged, the velites would chuck their javelins at them and retreat, goading them to charge straight through the channels. At the rear of the army, Scipio had units specially trained to fight elephants - and the elephants were dispatched, with more damage done to Hannibal's army than his own.

Funnily enough, the Romans adopted elephants to their own use, with moderate results. There are also reports of flaming pigs being used against elephants, but I don't have any sources to confirm this one ;)

Hope that answered your questions! If you have any more, please feel free to ask them!

Here are some ancient sources on how to deal with elephants throughout the Mediterranean. Generally speaking, they sought to spook and turn them around. Killing them was difficult and time consuming in a battle, and done mostly by missile weapons. I read about Legio V Alaudae taking down the elephants at Thapsus with axes, but I can't remember where.

"Down they came from their strong places, and hurling their javelins at the elephants compelled them to wheel about and run back through the ranks of their own men."

"Next, it was decided to run a trench parallel with the camp of the enemy, and at either end of it to set their waggons, sinking them to the wheel-hubs in the ground, in order that, thus firmly planted, they might impede the advance of the elephants. When they began to carry out this project, there came to them the women and maidens, some of them in their robes, with tunics girt close, and others in their tunics only, to help the elderly men in the work. 4 The men who were going to do the fighting the women ordered to keep quiet, and assuming their share of the task they completed with their own hands a third of the trench."

Plutarch, Life of Pyrrhus

"To complete their discomfiture the elephants, scared by the sudden onsets of the cavalry and the rapid movements of the light infantry, rushed from the wings into the centre of the line"

From the Founding of the City by Livy Book 28: The Final Conquest of Spain

9.6. Scipio drew up his army in the following fashion. 7. In front he placed the hastati with certain intervals between the maniples and behind them the principes, not placing their maniples, as is the usual Roman custom, opposite to the intervals separating those of the first line, but directly behind these latter at a certain distance owing to the large number of the enemy's elephants. 8. Last of all he placed the triarii. On his left wing he posted Gaius Laelius with the Italian horse, and on the right wing Massanissa with the whole of his Numidians. 9. The intervals of the first maniples he filled up with the cohorts of velites, ordering them to open the action, 10 and if they were forced back by the charge of the elephants to retire, those who had time to do so by the straight passages as far as the rear of the whole army, and those who were overtaken to right or left along the intervals between the lines.

Polybius Book 15, Chapters 9-19

33.1. Scipio now marshalled his troops for battle: in front the spearmen (hastati), and behind them the second rankers (principes), and then the third row men (triarii), closing up the rear. He did not deploy his cohorts in conventional close order in front of their individual standards; instead he organised them by maniples , with wide passages in between each, so that the enemy’s elephants would not disrupt the battle lines as they charged. 33.2. He put Laelius in command of the Italian cavalry on the left wing. He had been his deputy commander (legatus), but for the current year was a special quaestor, appointed by senatorial decree instead of by lot. Masinissa and the Numidians were on the opposite wing, on the right. 33.3. He filled the open passages between the maniples with platoons of skirmishers (velites), who were lightly armed in those days, and gave them strict orders to retreat behind the front lines as soon as the elephants charged, or else to scatter to left and right and link up with the front line troops, thus opening up a route for the elephants to charge through and leave them vulnerable to fire from both quarters.

Edit: I suck as formatting.

Livy Book 30, Chapters 32-36