You are in luck because we have what is essentially a guide to Roman political campaigns, the Commentariolum Petitionis ("A Pamphlet on Campaigning for Office" or something like that, each edition translates the title differently). The provenance of this source is a bit debated (it claims to be by Marcus Cicero's brother Quintus, but one wonders why Quintus seems to think he is in any position to give his older and more successful brother any advice) but most historians agree that it is a useful and accurate look at campaigns.

To answer your questions more specifically: there is a lot of scholarly debate about how the process worked. The old guard (see Lily Ross Taylor's Party Politics in the age of Caesar) argues essentially what you claim in the last part of your question: that Roman politicians built networks of powerful supporters who could mobilize vast numbers of clients to vote however they commanded. This is still the textbook narrative, but it is not how the Commentariolum nor any other Roman sources actually talks about winning an election.

Instead Cicero and his contemporaries talk about winning over public support by doing public deeds, highlighting previous public service and works, hosting large public events like feasts and games, and calling in every favor you have and offering to reciprocate any support you receive. Yes they did make campaign speeches and post fliers (there are political ads painted on the walls of Pompeii). Romans did not, as a rule, go in for official public debates. They didn't really discuss policy either, instead they highlighted their own reputations and tried to tarnish the reputations of their competitors. Many of these tactics were borderline illegal under bribery (ambitus to the Romans), so litigation was always a last-ditch and occasionally successful way to win (Cicero's Pro Murena is a defense speech on ambitus which gives another look at how Romans build their political careers).