Yes, the Romans certainly had a concept of technological progress, but one that is hardly comparable to ours. Here are some most obvious differences to consider; 1) the Romans did not always account technological advancement as much to the efforts of a line of human inventors, but saw it as a divine process which was influenced by gods and Nature; and 2) the Romans would not have separated 'technological' progress from other arts and science (literature, philosophy, theatre, poetry, astronomy...) as strictly as we do; and finally 3) the Romans did not see technological advancement as a purely positive thing: it had a corrupting effect on morals.

Ancient Romans, like all advanced societies, realized they had once been simpler. The Roman authors often examined the process by which their standard of living had improved. But, when historical sources are absent, logic, emotion, and religion become rational sources of explanation. Sometimes the ancients can assign a certain invention to a historical figure, but sometimes inventions (especially the most ancient and fundamental ones) are pictured as gifts from gods; e.g. Philostratus on painting:

For the person wishing to devise cleverly, the discovery of painting comes from the gods - observe that the Horai ['Seasons'] paint the meadows on the earth and the displays in the sky—but for the person seeking the origin of art, imitation is the eldest invention and the most related to nature; the wise invented it, calling it now painting, now plastic art.
Philostratus, Imagines 1.1 (294.5–12)

Elder Pliny, writing during the early Empire, devices a list of where all the different innovations of the army came from. As you can see, he does not separate the purely technological advancement from the abstract, and he indeed believes that truces and treaties were invented by one historical person. Also, although looks like he might be right on some things e.g. that ballistas came from the Phoenicians, we should probably be a bit sceptic about whether Mars’ son developed the spear or whether the Centaurs invented cavalry tactics…

The Africans were the first to use clubs — they call them “staves” —when they battled the Egyptians. Shields were invented, if not by Chalcus son of Athamas, then by Proetus and Aerisius while they were campaigning against each other. Midias of Messene invented the breastplate. The helmet, sword, and spear were inventions of the Spartans, and greaves and crests for helmets came from the Carians. Some say Jupiter’s son Scythes invented the bow and arrow, though others attribute the latter to Perseus’ son Perses. Lances were developed by the Aetolians, the spear with a throwing strap by Aetolus, Mars’ son, the light skirmishing spears by Tyrrenus, likewise the heavy javelin, the battle-axe by Penthesilea the Amazon. Pisaeus is credited with hunting spears and the version of missile-throwers called the scorpion, while the Cretans invented the catapult and the Phoenicians the ballista and sling. The bronze war-trumpet came from Tyrrenus’ son Pisaeus, the testudo from Artemon of Clazomenae, and, from Epius while at Troy, the style of siege-engine called the “horse,” now the “ram.” Bellerophon invented horse riding, Pelethronius reins and saddles, and the Centaurs — Thessalians who lived beside Mt. Peleus — cavalry tactics. The Phrygian race was the first to harness two horses to a chariot, and it was Erichthonius who added two more. During the Trojan War Palamedes invented military formation, the password, tokens for recognition, and sentinels; Sinon in the same war invented signalling from watchtowers. Truces and treaties were invented by Lycaon and Theseus respectively.
the Elder Pliny, Natural History 7.200–2

The ancients believed that the inspiration for technological advancement came from the Nature. The gods had given people the skills of deliberation, speech, social organization etc. over other animals, and these skills produced strife, jealousy, and rivalry that resulted in technological and economic innovations, as humans strived to find tools to rise above animals and each other. Nature was thus the spark that drove people to build and invent; some ancients, like the Roman Republican architect Vitruvius, also believed that even the most artificial mechanisms could also ultimately be found in Nature, and human invention was simply imitation of Nature:

All mechanisms are created by nature and founded on the revolution of the universe, our guide and teacher. For example let us first contemplate and examine the continuous motion of the sun, moon, and the five planets. Unless they revolved by natural means, we would not have had alternating light [and dark] nor would fruits ripen. Thus when our ancestors had realized that this was so, they took their examples from nature, and by imitating natural examples they were borne onward by divine truths which they adapted to their way of living. As a result they discovered that some things were more easily done with machines [machina] and their revolutions, some others with instruments [organum]. Thus they took care to improve gradually by their learning all those things which they believed useful for research, for the arts, and for established traditions.
Vitruvius, On Architecture 10.1.4–6

This is not the case with all ancients though, and e.g. Aristotle had in Classical Athens specifically seen machines and devices as a sign of people rising ABOVE nature; Vitruvius is probably influenced by the Hellenistic school of Stoic philosophy. But, both Aristotle and Vitruvius believed that because nature was the main source for human advancement, climate and geographical location had an effect on the progress of a society. Aristotle says that cold climates are "lacking in intelligence and art" but have too much spirit, reducing their people to a level of impulsive barbarians, whereas Greece is ideally located to have all the spirit, intelligence and art. The Roman architect Vitruvius similarly believed that Italy was ideally located for an advanced, civilized and innovative people.

Consequently, the Romans did not really use Greece (as the OP mentioned ancient Troy) as a measuring stick for their technological advancement. After all, the Greeks were also a civilized and sophisticated people who possessed humanitas, and all Roman intellectuals read Greek texts. There are passages that say something to the effect ‘we Romans build baths and aqueducts and roads which the Greeks did not’, but it does not make the Romans superior to the Greeks. Again and again Roman writers like Cato, Columella, Frontinus, and Pliny the Elder stress the practicality of Roman culture over the Greeks that manifests in Roman agriculture, administration, and military, but because the Romans considered the Greeks more advanced in their language, literature, and arts, the Roman did not see themselves as more advanced than the Greeks. The Romans did, however, spent a lot of time comparing themselves to the 'simple and unadvanced barbarians' around them, and very much believed that civilizing other peoples was part of their imperial enterprise; not always to a positive effect, as Tacitus believes (he’s talking about his father-in-law’s campaign in Britain):

The following winter [A.D. 79] was spent in very beneficial consultations. For in order that the scattered and barbaric Britons, a people ready for war, might be accustomed to pleasure by means of peace and relaxation, Agricola, by praising the enthusiastic and scolding the lazy, urged on individuals and assisted communities to construct temples, market places, and homes… Gradually the Britons yielded to the enticing vices: the covered porticoes, the baths, and the elegance of banquets. And this condition was called “civilization” among simple Britons, although it was part of their slavery.
Tacitus, Agricola 21

This Tacitean passage reveals another Roman attitude towards technological progress; it is in fact not always good. The Romans believed that long time ago, their people had lead a very primitive existence of early humans, a time in which society was too simple to create its own technological inventions, where life was all about family, simple pastoral and strictly religious life style, and war. These earlier Romans were also morally superior to the contemporary Romans. Baths, games, luxuries, riches, and machines had corrupted the virtues of family, piety, respect, frugality, and masculinity that the Roman society had once had. The Romans loved dwelling in this sort of golden-hazed nostalgia towards their imagined, mythohistorical past. The Romans placed the historical Troy and Trojan wars to this period of heroic and glorious past. So, although the Romans might have recognized that they now had weapons that the archaic Greeks did not have, they would have never considered their society superior to that of the Troy of ancient heroes. The Romans did not see technological advancement as the the most important criterion for the well-being and sophistication of a society the same way that we moderns might do.

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I sort of have a follow up, was there any sort of military research or weapons research? Did you have people researching the optimal size of a legionnaires shield, someone researching the optimal pillum weight? Or was it mostly, Titus the centurion has come up with a better type of helmet for him and his men, and then everyone else is like hey that works pretty well let's copy that.

There is a great article on this subject by Fernando Echeverría in a volume titled New Perspectives on Ancient Warfare (2009). His starting point is that most modern theories as to why one ancient people defeated another are technologically determinist: the Greeks beat the Persians because they had better weapons, the Macedonians beat the Greeks because they had better weapons, the Romans beat the Macedonians because they had better weapons. While there is some limited justification for this in the sources, the way the ancients looked at this was actually quite different.

Echeverría's main point is that ancient peoples did not have a concept of progress. They did not conceive of the world as marching steadily toward a better future through discoveries in science, engineering and the like. They recognised change, of course, but they would not have recognised our belief in, and striving towards, constant improvement.

As a result, while the ancients obviously realised that their weapons were different from those of older peoples, they would not necessarily have regarded their weapons as better. The Romans created a narrative for themselves in which they learned new military techniques as they became appropriate. They learned to fight as heavy infantry in their wars against the Etruscans; they learned to fight in maniples in the rugged land of the Samnites; they learned to fight at sea in the First Punic War. The important point is that these developments were not regarded as progress but as the acquisition of a greater toolbox, with each tool having a particular purpose within particular circumstances. In the same way, they would have acknowledged that the weapons of ancient peoples were appropriate to their particular situation, not inferior or technologically backward.

The best proof of this, of course, is the fact that the Romans themselves adopted the weapons and fighting methods of others to face particular challenges. Against the horsemen of the east, they increasingly adopted spears and missile weapons, which had of course been the weaponry of the Greeks and Macedonians before them. Due to constant conflicts with Central European peoples, they eventually made heavy cavalry the core of their armies. Only an evolutionary notion of military development would regard such steps as inevitable "progress"; the Romans merely saw them as effective adaptation to circumstance.

Edit: posted at the same time as the much more comprehensive intellectual history provided by /u/mythoplokos. I don't think we are in disagreement, though - my post is more or less complementary to his/hers.

I have a few follow up questions to do with presentism.

Did more Romans value artistic progress or technological progress? Did they see a difference? So, would they have seen something like patterns on armour or advancements in metallurgy as artistic or technological progress?

Additionally, would their mark of technological advancement be a well made coin or a pure (un-debased) coin?

Archimedes was recorded by Polybius after building Sambucae (ladder boats) to defeat fortifications during an amphibious landing in Sicily: "So true it is that the solitary intelligence of one man can be a miracle when (properly) applied… " Now, Archimedes was not Roman, but the Romans certainly recognized the value of technological advancement and progress, and were as such intensely aware of variations in tactics as well as innovations in siegecraft, infantry combat, and especially logistics and supply.

Source: Perseus database at tufts.edu (publicly available archives)