The Romans took their road construction seriously, as evidenced by the fact that many of them still exist and some are still in use, two thousand or more years after they were first laid down. The expansion of the Roman road network paralleled the expansion of Rome itself, as the roads were crucial for the movement of trade goods, messengers, and most crucially, soldiers.

The foundation of the Roman system of roads goes all the way back to the Laws of the Twelve Tables, released in 449 BC and considered to among the founding laws of the Roman Republic. These laws mandated a minimum width for roads, who had the right to use them, and the responsibility of private property owners to allow public roads on their lands if deemed the proper route. The first major Roman road was the Via Appia (Appian Way), which linked Rome and the city of Tarentum in the south (later extended to the Adriatic port of Brundisium). The first section opened to traffic in 312 BC, and the remainder was completed in 264, even running a course of 212 km. It was built in response to the great difficulty Rome had moving troops south to fight the first Samnite War. By the end of the third century BC, other roads (the Via Aurelia, Via Flaminia, Via Latina, Via Aemilia, and Via Valeria) had been built, and it was possible to reach most of Italy on a well-built road.

And well-built they were. Roman strategy was to make them right the first time and worry less about maintenance later. Most people are familiar with the surface of the Romans roads – a network of large flat stones fitted loosely together – but this is indeed just the surface.

Roman road construction started with the digging of a trench a meter or more deep down to the foundation soil. After this was reached the road was constructed in layers.

1. The bottom layer was the foundation soil itself, compacted to be hard and even and covered with a layer of sand
2. The next layer was the statumen, made of fist-sized rocks to a depth of 30-60 cm.
3. The rudus was a layer of crushed rock a few cm in size and bound with mortar, around 20 cm thick
4. The nucleus came next, and was a mix of sand, gravel, and cement 30 cm in depth.
5. The summum dorsum is the top, visible layer made of flat blocks of around 15 cm in thickness.

The roads were around 6 m wide to allow two way traffic of carts and troops. They were cambered higher in the middle to allow drainage down to the curb. There was often a strip of packed earth or gravel specifically for pedestrian and horse traffic to the side of the road. The resulting roads were durable and all-weather capable. The Latin word for layers is strata, from which we derived the word street.

Road construction occurred on an ad hoc basis. Basically, as the Republic (and later the Empire) expanded, roads were built to the newly-conquered territories. Generals who conquered new lands were expected to build roads into them at their own expense (financed but the plunder of their conquests). Most of Roman expansion was done by the end of the second century BC. A second wave of construction accompanied the move of the Roman capital to Constantinople in AD 323, as the city was essentially new and required a new infrastructure to support it.

So to answer your question, the total road network took as long to build as the empire lasted (all the way to 1453 in the East). But the major parts of the network were already in place by 100 BC, and had taken over 250 years to get that far.

Further reading:

Roads and Bridges of the Roman Empire, Horst Barow, 2013