First thing to consider: What we now know as an engineer didn't begin to form until the late Renaissance , the need to ground the practice on science. Before that what we consider classical engineering was for the most part based on empirical knowledge. An exception could be made for the simple machines that were studied in the times of Archimedes. Most of what engineering is based on could not be solved until we had an understanding of calculus that did not happen until the 17th century.

What we consider classical engineers could have origin as diverse as a barber or a medic. The work of the artisan often depended of what the Patron wanted. Many times the best artisans found themselves working at the whims of whoever paid them, finding versatility essential. Some engineers were highly skilled master craftsmen, others were simply civil servants that for some reason found themselves in charge of departments dealing with engineering matters. In the end an engineer was simply an artisan that had reached the top of his profession and that has attained responsibilities beyond what a simple artisan would charged with.

So to answer the question, the best bet to become an engineer would be to either become an artisan and go up in the ranks (mason, metalworker, carpenter) or to find yourself in the employment of a patron that needs the solution of a problem.

Sources: A History of Engineering in Classical and Medieval Times (1984) by Donald Hill

Engineers and Engineering History: problems and perspectives (2004) by Antoine Picon

There were two or more Roman centuriae of fabri, artisans, divided into groups by trade along the lines of medieval guilds. We know virtually nothing about them except that they existed. They were supposedly institutionalized in the constitution of Servius Tullius, early Rome's quasi-mythic 6th king. Livy (1.43) says that there were two centuries total and that they voted with the first class -- that is, with the upper echelon as far as socio-economic status. They did not carry weapons but were assigned the duty to construct siege machines (ut machinas in bello ferrent). Cicero, for what it's worth, only mentions one centuria of fabri (Rep. 2.22). We hear of fabri tignarii (carpenters), fabri aerarii (smiths), and sometimes fabri ferrarii (either smiths again, or perhaps weaponsmiths specifically). We know nothing about how they operated as a corporate group, if they even did at all. It is likely that they maintained knowledge of engineering tasks (like building bridges) and passed that ability down through generations by apprenticeship, just like guilds, but there is no evidence or mention of it to my knowledge. It is possible they had a school (like a modern trade school), but we hear of nothing like that. We know that Roman legions were capable of engineering feats with little or no prior warning, so it follows that there was some manner of bringing this know-how along on the march. Not every Roman had the knowledge of bridge-building, surely. Caesar, for instance, decided to build a bridge across the Rhine and attack the Germanic tribes on the other side during his adventures in Gaul in 55 BCE (BG 4.17-19). It was done quickly and effectively, and Caesar (tries to) describe the manner of construction. It is clear that even he is impressed, and wants to write to Rome about how impressive this thing he is doing really is. Somebody in those legions had been trained in engineering, but we don't know how that worked at all.

The most obvious candidate for engineering know-how in the legion was the enigmatic post called the praefectus fabrum (chief of the artisans). The post is mentioned quite often by Roman writers but its function is quite unclear in any period. Vegetius says there was one with every legion, but there is no inscriptional evidence of praefecti fabrum in association with individual legions. I think it is likely that the office was attached to the consul (or praetor or proconsul or propraetor or legatus or whoever was in charge of the army in question). It would be logical that this person be drawn from one of these centuriae fabrum but again, there is no evidence to support the idea. Some praefecti fabrum in the Late Republic seem to have held the post for political, rather than knowledge, reasons. Caesar's P.F. in Gaul was Mamurra, from Formiae and of equestrian rank. Catullus mentions him in a few poems (he was screwing Mamurra's girlfriend, if I recall correctly; he also calls him mentula 'dick'). We have no idea what the job of the P.F. actually was, but it follows that he managed a contingent of fabri based on the name alone, and was probably expected to carry out whatever engineering task needed doing (bridges, ship-building, trench-works, and the kitchen sink are all required by Caesar in Gaul -- it must have been a trying job). On most tasks, they had to have engineering as well as organizational skills to manage skilled labor (the fabri) and also literal legions of unskilled labor (the rank and file soldiers who did most of the digging and chopping and hammering, most likely).

To complicate matters, there was also one or more praefectus fabrum in Rome (sometimes with Romae 'at/in Rome' in inscriptions). These were clearly separate from the army prefects, but their origin and general duties were probably similar in the beginning. By the Imperial period, we find the emperor doling out mundane administrative tasks to them. They may have worked for the aediles (or later Emperor) in some technical capacity as civic structures were built or repaired.

I think the best place to check out the P.F. is Welch, "The Office of the Praefectus Fabrum in the Late Republic," Chiron 25 (1995): 131-45.

Additional follow up: For what social ranks would this be considered a respectable profession? Not a patrician, presumably, but what about a member of the Equites? A son of a wealthy plebeian? A son of a poor plebeian?

Follow up question. Were these enormous projects built by some form of Corps of Engineers, or was it outsourced to tradesmen?