I'm going to pass on the conflict of the orders. I'm not at all convinced we can give a good account wie es eigentlich gewesen for anything before the Punic Wars, and repeating the traditional account as we get it in our Augustan sources won't get us more than, well, the traditional account as we get it in our Augustan sources. You seem to be fuzzy about the institutions of Roman government, though, and I can help with that.

Note the Roman constitution was an evolving thing for the 500 years of the Republic. What I'm describing here is the developed form, more or less, it had at the end.

The Centuriate Assembly (what you call the Soldier Assembly I think) war primarily responsible for military affairs; the election of magistrates with imperium, declarations of war, trying capital crimes (people often forget that consuls' primary job for most of the Republic was the commanding of armies - concerns internal to the city were secondary). This committee met in what were intended to be groups representing various levels of military commitment to the State, although by the Late Republic the form persisted despite the failure of Rome's wealth-based military enrollment.

The Tribal Assembly elected minor magistrates, handled minor trials, and passed laws. Tribes represented geographical divisions, the most important being the difference between the urban tribes (people living in the city) and the rural ones (people living outside the city). Strict geographical boundaries of tribes also were falling apart in the 1st century BCE.

Closely associated with this is the Plebeian Assembly, which also could pass laws (plebicites rather than leges) and elect Plebeian tribunes.

The Curiate Assembly was traditionally the oldest, but over time had been relegated more and more to religious functions.

The Senate is a different animal altogether. It traditionally advised magistrates (including advice on what legislation should be brought to the people for voting on), handled foreign diplomats, and state finances. The Senate technically had little to no coercive power. It expressed itself in opinions ("senatus consultum" means an "opinion of the Senate"). Magistrates took Senatorial supervision more or less seriously; there's a case from Cicero's tenure of governor of Cillicia in which he feels compelled to enforce the collection of a loan Brutus made to one of the local municipalities. Cicero thought the terms of the loan were a scandal but in the end when the Senate issued an opinion that the terms should be enforced, he did, although he lawyered his way to a much better deal for the city (nb, don't take loans from Roman senators).

This is all an approximation, and exceptions to this neat little package of Roman government happened with what strikes me at least as disturbing regularity. Several times we hear of the Centuriate Assembly demanding people disqualified from office for one reason or another, and getting its way. Ti. Gracchus got his hands on funds for his settlement program despite the supposed Senatorial control of the purse, etc. Romans were really good at fudging things if it was convenient.