The Senate was an advisory body whose decrees (senatus consulta) were not laws (leges). As such, a magistrate had to call the Senate to meet in order to ask its advice. This was regularly done three times a month, on the Kalends, Nones, and Ides (hence Julius Caesar is killed in the senate on the Ides of March). Extraordinary meetings could also be called. Meetings had to take place within Rome or her immediate environs and within a space which had been consecrated by the augurs (a type of priest). The curia, or senate house, was the usual spot, but in practice any temple could be used. A quorum was necessary, but since the number of senators varied over time, presumably the number necessary for a quorum did too.
The presiding magistrate (usually one of the consuls or, if they should both be absent, a praetor) sacrificed and took auspices before the session formally began. If the omens were bad or the ceremony done improperly, the session was postponed.
The presiding magistrate then opened the session with a speech, asking the senate's opinion on some matter or another. There was a set order for who got to speak first, though that order seems to have varied over time. Originally the princeps senatus, a particularly old and distinguished senator, spoke first. In the Late Republic the order of precedence might have been determined by offices held, so those who had attained the rank of consul spoke first, then those who had been praetor, and so on, down to the most junior man in the room.
Regardless of the precise order of precedence, the point was to allow older men to speak first, and so allow their position to have the most influence. If five ex-consuls have said more or less the same thing, its hard as an ex-praetor speaking after them to disagree. The system was designed to generate consensus, not to spur on endless debate.
If voting was necessary, it was done by physically moving near to the senator you support. So when Cicero delivered his first speech against Catiline, we are told that the senators gradually moved away from Catiline until Catiline was left practically alone.