Latin Military fundamentally divides into

1. Troops that require an oligarchic government.

2. Troops that require a democratic government.

3. Troops that don't care about the government.

4. Socii that can be trained from NON-Latin Settlements. But are unique to Latins.

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1. - The Latin League.

Rome was not the only Latin Settlement but by far the most influencial, probably even around 500 bc. Unlike Rome however that had just gotten rid of their red socked King and established a Republic(well thats the legend anyway) not much is "known" about the politics of other Latin Communities. And the Romans themselves are also pretty new to the Republican System. Thus you will find that other Latin Settments do not actually grant Republican troops from the get-go but Tribal/oligarchic troops:

Famuli - cheap skirmishers
Miles - decent medium swordsmen (requires Armoury)
Nobiles - Mighty Elite Hoplites (requires Established Nobility)

In order to get those Republican ones you will have to research "Res Publica" and build a democratic government first.

2. - The Republican Infantry

2. 1. - the ("Servian")Hoplite Army

These Guys are where it's at, proper Roman heavy infantry, with Hastati and Principes. But wait! this is Hegemony 500 not 400(or 350), so none of that for you! Well not yet. First we have the often mentioned "Hoplite Army". It's essentially the same as the Etruscan Democratic System. the really rich dudes on horses the rich ones as heavy hoplites in front and the others sorted by decending wealth behind them.

Leves - cheap Skirmishers (get replaced by Velites)
Roarii - weak spearmen (get replaced by Velites)
Tertiani - ok light infantry with both melee and ranged capacity
Secundi - Light Hoplites (requires Armoury)
Classici - Heavy Hoplites (requires Armoury and Established Nobility)

2. 2. - The "Livian"/"Camillian"/"census-based manipular-" Army

Now to what all of you have been waiting for, you know them, you love them and no matter what you call them they are the first glimpse at Romes famous Pragamtic Military Doctrine and it's posterboy: the ("marian")Legionary. But let's not get ahead of ourselves.
In order to train these boys, you need the "Social Reform" Skill(it's in the Social Tree in case you were wondering). This represents the important shift away from placing the best troops at the front and now using them as reserves for when your enemy has chewed through all of your crappy troops. (on details, just text me, but this is getting long as it is) In contrast all other major groups of the era basically retained the idea of engaging primiarily with the best troops only the germanic tribes did something similar and they still have to gather some food to press that age up button.

Hastati
Principes (requires Armoury)
Triarii (requires Armoury and Established Nobility)

2.3. - The "Polybian"/"experience-based manipular-" Army

Polybius is not the guy who instated these reforms, he just described them a lot. They are the Legions of the Second Punic war. The balance is shifted towards the Hastati and Principes which are much more effective due to using heavier equipment and being fielded in greater numbers. Sadly however the Triarii are quite a bit weaker now. tbh I would just use regional or Socii hoplites, those are better. but if you want to roleplay...
To get them you need to research "Military reforms" skill. You also need the Social Reforms, otherwise you'd only get the Velites.

Velites (replaces Accensi, Leves and Roarii)
Hastati (now better)
Principes (now better, requires Armoury)
Triarii (now worse, requires Armoury)

3. - Other latin Troops

These don't care about the Government and have their own requirements, mainly the tech Extended Citizenship in the Social Skilltree:

Accensi - dirt cheap slingers (requires Extended Citizenship, is replaced by Velites)
Equites - Cavalry (requires Established Nobility and Warhorse Studs)
Equites Publico - slightly different Cavalry (requires Warhorse Studs, Extended Citizenship and Social Reform)

4. - Socii, Italic Allies

Honestly, just look around, they generally give you 1-2 Brigades that will be displayed before other local troops. These tend to be a bit heavier than the Original. Once you unlock the "Socii" Skill you can train them from Sabellic, Etruscan and Greek settlements.
Just keep in mind that "Velites Sociorum" can be trained in both greek and sabellic settlements but get replaced by heavier troops if you have an Armoury. More are coming:

Velites Sociorum - Javelin infantry
Hastati Sociorum - Light Infantry
Pedites Grecorum - Hoplites
Pedites Tyrrhenum - Noble Hoplites
Equites Extraordinarii - Noble Lancers
Pedites Extraordinarii - Elite Infantry