The Senate was tightly controlled by the Roman emperors, so to answer this question we’ll concentrate on the Roman Republic. The Senate was the supreme council of the Roman state and anyone with any importance would have belonged to it.
Basically entry depended on having been elected to one of the annually appointed posts in the Roman government and administration, eg the consulship.
Normally a few years would have gone by before a man could expect to join. Membership was for the most important men in Rome from well-to-do families who possessed the most wealth and influence within this community.
Membership was for life unless expelled for misbehaviour. It was the annually elected magistrates who ran the government but the Senate gave formal advice and it was up to a consul to enforce it.
During the Second Punic War (which took place from 218-201 BCE) the Senate made crucial strategic decisions and handled the war-effort, which essentially led to the defeat of Hannibal and Carthage. The little-remembered Second Punic War is one of the most famous victories in the history of ancient Roman warfare.