It's pretty simple, actually. Get elected to office. Any office on the cursus honorum would have got you in once your term expired.

There are exceptions of course - people being elected before they were eligible happened occasionally. Pompey famously entered the Senate after his first election, to the consulship! And towards the end of the Republic dictators starting appointing people to the senate - they were dying off too fast to maintain the small bureaucracy Rome had for managing the empire.

Election would mean you needed connections with established political families to support you, and money. Your occupation didn't matter so much. Cicero did it by being a lawyer, but Marius by being a soldier. Cato the elder also was a soldier. Antony accused Octavian of being the son of a muleteer.

I'm not sure there's much else to say on the matter. The senate was strictly speaking a collection of former magistrates that advised current magistrates. Due to the way the cursus honorum worked (holding successive offices for a year at a time) most of them could expect to become magistrates again.

There are a shockingly low number of new men on record though. Strictly speaking that term refers to people who achieved the consulship for the first time in their family history. If I counted right, Wiseman in New Men in the Roman Senate lists 52 between 139 bce and 14 ce, and almost all of these were suffect, meaning they served only part of a term.