Initially the Roman army consisted of milicians,
who were selected for one campain

Though, until the reforms of Gaius Marius, Rome had no standing professional army, the army always hs been the core of the Roman society. No political carreer was possible without having served in the army

King Servius Tulius, according to Titus Livius, reorganized the Roman army and created the taxing system (census) on which the position and armament of each man was determened.
Polybius 6.12:

As soon as the consuls are declared, the military tribunes are next appointed. Of these, fourteen are taken from the citizens who have carried arms in five campaigns; and ten more from those who completed ten. For every citizen, before he arrives at the age of forty-six, is obliged to serve either ten years in the cavalry, or sixteen in the infantry: those alone excepted who are placed by the censors below the rate of four hundred drachmae; and who are all reserved for the service of the sea. In the case of any pressing danger the time of continuing in the infantry is extended to twenty years. No citizen is permitted by the laws to sue for any magistracy before he has completed the serving of ten campaigns.
When the enrollments are to be made the consuls give notice before to the people of a certain day, upon which all the Romans that are of sufficient age are required to attend. This is done every year. And when the day arrives, and the men all appear at Rome, and are assembled afterwards in the Capitol, the tribunes of the youngest order divide themselves, as they are appointed either by the consuls or the people, into four separate bodies. For this division corresponds with the first and general distribution of all the forces into four separate legions. Of these tribunes, therefore, the four first named are assigned to the first legion; the three next to the second; the
following four to the third; and the last three appointed to the fourth. Of the tribunes of the oldest order the two that are first named are placed in the first legion; the three second in the second; the two that follow in the third; and the remaining three in the fourth. By this distribution and division an equal number of commanders is allotted to each legion.
When this is done, the tribunes of each legion, having taken their seats apart, draw out the tribes one by one by lot; and calling to them that upon which the lot first falls, they select from it four young men, as nearly equal as is possible in age and stature. And when these are brought forward from the rest, the tribunes of the first legion first choose one; then those of the second a second; those of the third take the third; and those of the fourth the last. After these four more are made to approach. And now the tribunes of the second legion first make their choice; then those of the rest in order; and last of all the tribunes of the first. In the same manner again, from the next four that follow, the tribunes of the third legion choose the first; and those of the second the last. And thus, by observing the same method of rotation to the end, it happens that the legions, with respect to the men of which they are composed are all alike and equal. The number allotted to each legion is four thousand and two hundred; and sometimes five thousand, when any great and unusual danger is foreseen. After these had been thus selected it was anciently the custom to choose the cavalry; and to add two hundred horsemen to each four thousand of the infantry But in the present times, the citizens, of whom the cavalry is composed, are first enrolled; having been before appointed by the censors, according to the rate of their revenue; and three hundred are assigned to every legion.

Of course the armament of this army and the way it would operate was considerably different from the imperial period. Since there are 700 years in between, this is to be expected.  The Romans didn't mind copying idees from other peoples, as long as it improved the Roman army, they would adapt.

The base of the Roman army was of course the legionaire. More than anything else the army depended on the infantry. The cavalry was mainly there to protect the flanks in battle and reconnaissance For rich young men, wanting a political career, the cavalry was the opportunity to make themselves noticed.
These legionaires together formed the legion.

From the time Rome had forced the conquered cities to be allies of Rome, they had also to produce soldiers. Most of the cavalry was delivered by the allies.

In the republican era a consular army existed of two Roman legions in the centre and at each wing an ala with allied soldiers commanded by a Roman. An Ala was organised in a simalar way as the Roman legion.