Well... if we are to go with what Tolkien put in the books he is wearing a dark green cloak and boots.... oh and there is mention of a pipe and a sword at his side so that would mean a belt but otherwise no mention of any other clothing.

These are all of the descriptions of Aragorn that I can find -

Suddenly Frodo noticed that a strange--looking weather beaten man, sitting in the shadows near the wall, was also listening intently to the hobbit talk.He has a tall tankard in front of him, and was smoking a long stemmed pipe curiously carved. His legs were streched out before him, showing high boots of supple leather that fitted him well, but had seen much wear and were now caked with mud. A travel stained cloak of heavy green cloth was drawn close about him, and in spite of the heat of the room he wore a hood that overshadowed his face; but a the gleam of his eyes could be seen as he watched the hobbits.

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"As Frodo drew near he threw back his hood, showing a shaggy head of dark hair flecked with grey , and in a pale stern face a pair of keen grey eyes.

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You may meet a friend of mine on the Road: a Man, lean, dark, tall, by some called Strider.

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He stood up, and seemed suddenly to grow taller. In his eye gleamed a light. keen and commanding. Throwing back his cloak, he laid his hand on the hilt of a sword the had hung concealed by his side.

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Aragorn was the tallest of the Company, but Boromir, little less in height, was broader and heavier in build.

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For the grim years were removed from the face of Aragorn, and he seemed clothed in white, a young lord tall and fair;

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Then Aragorn took the stone and pinned the brooch upon his breast, and those who saw him wondered; for they had not marked before how tall and kingly he stood, and it seemed to them that many years of toil had fallen from his shoulders.

Maybe other will be able to come up with more quotes that may shed light of the clothing issue!

However if you just think about it in a logical way he most probably wore breeches and a tunic as well as a vest of some sort. Probably in leather and heavy wool to keep warm.

In terms of literary and historic background at least, it is fairly clear that Tolkien based most of the human races in Middle-earth on Anglo-Saxon and Norse models. It is equally clear that the climate and geography of the areas that that story mostly covered are similar to northern Europe. It is hinted that the south of Gondor and the environs of Minas Tirith are a bit more Mediterranean but as the question is about Aragorn, his role as a ranger would tend to place his dress at the start of the books in a more northern mode.

Most historical sources place the inhabitants of northern Europe firmly in the trouser wearing or full length robe camp. Given that Aragorn and companions hop onto horses and hike through swamps, trousers seem more likely.

There is also another piece of evidence from the text in 'The Tower of Cirith Ungol' p. 892 (2001 edition) just after Sam rescues Frodo we have the lines:

He opened the bundle. Frodo looked in disgust at the contents, but there was nothing for it: he had to put the things on, or go naked. There were long hairy breeches of some unclean beast-fell, and a tunic of dirty leather. He drew them on ...' (my emphasis).

This certainly gives us evidence in-narrative that trousers of some sort were known technology to the Hobbits at least. Consider that they are called 'breeches' rather than strange orcish leg coverings. This sort of implies that trousers of some sort were known.

You also have to sort of assume that the 'born in the saddle' Rohirrim would have worked out the advantages of trousers at some early stage in their domestication of horses.

Apart from anything else simple common sense would suggest that anyone traveling in this sort of climate would opt for some sort of leg covering, call it breeches, hose, trousers, leggings or whatever.

We also have to consider that Tolkien was writing in the early 20th century when the basic idea that active men wore trousers outdoors or some medieval equivalent as a matter of course and so it's not unreasonable to assume that any special Middle-earth deviation from this would have been noted if he thought it was important.

Hobbits wore breeches. In addition to the two quotes mentioned in the other answers, here are three more:

In the section where Frodo shows Gandalf the ring the first time:

Frodo took it from his breeches-pocket, where it was clasped to a chain that hung from his belt. He unfastened it and handed it slowly to the wizard. It felt suddenly very heavy, as if either it or Frodo himself was in some way reluctant for Gandalf to touch it.

And in the Tom Bombadil section:

‘Dressed up like this, sir?’ said Sam. ‘Where are my clothes?’ He flung his circlet, belt, and rings on the grass, and looked round helplessly, as if he expected to find his cloak, jacket, and breeches, and other hobbit-garments lying somewhere to hand.

And where Frodo takes Sting after meeting Bilbo in Rivendell:

‘Very well, I will take it,’ said Frodo. Bilbo put it on him, and fastened Sting upon the glittering belt; and then Frodo put over the top his old weather-stained breeches, tunic, and jacket.

I did also find a few uses of "trouser-pocket", though no "trousers":

At the start of Bilbo's birthday speech:

They could all see him standing, waving one hand in the air, the other was in his trouser-pocket.

Merry speaking, talking about Bilbo:

"...I caught a glint of gold as he put something back in his trouser-pocket."

But I found no uses of "pants", "chaps", "overalls", "pantaloons", "leggings" ... Tolkien really wasn't interested in clothing, I guess.

TL;DR: There is good reason to believe that he wore pants, and no reason to believe that he didn't.

I searched all three books for relevant terms (e.g., pants, breeches, chaps, hose, leg, legging, knee, hip, waist, boot, sock, stocking, trousers, belt, pocket, pouch, leather, wool, linen, cloth, clout, etc), and picked out the ones that refer to Aragorn.

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Fellowship of the Ring:

Suddenly Frodo noticed that a strange-looking weather- beaten man, sitting in the shadows near the wall, was also listening intently to the hobbit-talk. He had a tall tankard in front of him, and was smoking a long-stemmed pipe curiously carved. His legs were stretched out before him, showing high boots of supple leather that fitted him well, but had seen much wear and were now caked with mud. A travel-stained cloak of heavy dark-green cloth was drawn close about him, and in spite of the heat of the room he wore a hood that overshadowed his face; but the gleam of his eyes could be seen as he watched the hobbits.

High boots on a man generally accompany pants

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The ground now became damp, and in places boggy and here and there they came upon pools, and wide stretches of reeds and rushes filled with the warbling of little hidden birds. They had to pick their way carefully to keep both dry-footed and on their proper course. At first they made fair progress, but as they went on, their passage became slower and more dangerous. The marshes were bewildering and treacherous, and there was no permanent trail even for Rangers to find through their shifting quagmires. The flies began to torment them, and the air was full of clouds of tiny midges that crept up their sleeves and breeches and into their hair.

Presumably, Aragorn's breeches are included

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He [Aragorn] sat down on the ground, and taking the dagger-hilt laid it on his knees, and he sang over it a slow song in a strange tongue. Then setting it aside, he turned to Frodo and in a soft tone spoke words the others could not catch. From the pouch at his belt he drew out the long leaves of a plant.

Usually, belts accompany pants

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Aragorn had Anduril, but no other weapon, and he went forth clad only in rusty green and brown, as a Ranger in the wilderness.

He has only one weapon - a broken and mostly useless sword - so the belt is probably for holding his pants up

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With that he sprang forth nimbly, and then Frodo noticed as if for the first time, though he had long known it, that the Elf had no boots, but wore only light shoes, as he always did, and his feet made little imprint in the snow.

Frodo finds Legolas' meager footwear remarkable - wouldn't he comment on Aragorn not wearing pants?

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They were so deep in the doings of the Four Farthings that they did not notice the arrival of a man clad in dark green cloth. For many minutes he stood looking down at them with a smile.

His pants are presumably dark green

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Elrond was there, and several others were seated in silence about him. Frodo saw Glorfindel and Glóin; and in a corner alone Strider was sitting, clad in his old travel-worn clothes again.

Pants are an important part of "clothes"

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The first part of their journey was hard and dreary, and Frodo remembered little of it, save the wind. For many sunless days an icy blast came from the Mountains in the east, and no garment seemed able to keep out its searching fingers. Though the Company was well clad, they seldom felt warm, either moving or at rest.

Well clad people usually have pants on.

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The Two Towers:

Slowly Aragorn unbuckled his belt and himself set his sword upright against the wall. ‘Here I set it,’ he said; ‘but I command you not to touch it, nor to permit any other to lay hand on it. In this Elvish sheath dwells the Blade that was Broken and has been made again. Telchar first wrought it in the deeps of time. Death shall come to any man that draws Elendil’s sword save Elendil’s heir.’

Again, a belt is usually paired with pants, but now the sword is fixed, so it is probably just a sword belt

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‘Here are some treasures that you let fall,’ said Aragorn. ‘You will be glad to have them back.’ He loosened his belt from under his cloak, and took from it the two sheathed knives.

This sounds like a separate, pants-holding-up belt

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‘Now let us take our ease here for a little!’ said Aragorn. ‘We will sit on the edge of ruin and talk, as Gandalf says, while he is busy elsewhere. I feel a weariness such as I have seldom felt before.’ He wrapped his grey cloak about him, hiding his mail-shirt, and stretched out his long legs. Then he lay back and sent from his lips a thin stream of smoke.

Mail shirt sans pants would be uncomfortable

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For many hours they rode on through the meads and river- lands. Often the grass was so high that it reached above the knees of the riders, and their steeds seemed to be swimming in a grey-green sea.

His knees would be cut to ribbons without pants here

I can't imagine the Rangers of the North not wearing pants/breeches or trousers of some sort. They were a warrior group whose vocation, we might guess, would have involved a lot of horsemanship and riding. Given the historical connection between horse-riding cultures and the development of trousers, I just think it is more likely that at least in his capacity as a Numenorean Ranger, Aragorn must have worn pants/breeches.

Tolkien also based the cultures in Middle Earth (especially of Men in the northern regions) on Germanic societies. This too seems to favour the use of pants/breeches over any tunic-like apparels more common in Mediterranean cultures. However, once he took up his role as High King of the reunited Gondor and Arnor, Aragorn may have switched to the long flowing robes of elf-like apparel. We just don't have enough information to say anything definitive.