One thing I keep saying about Cerberus' power level in the trilogy timeline:
Cerberus is written to always have things going on that we're barely aware of until its in front of us.
They're a small rogue faction in ME1. Sure, okay. Lets ignore that they seem to have their paws on a lot of material/organisms that at least require inside-men in several human corporations.
They're a terrorist group with Alliance and corporate ties in ME2. Sure, okay. Lets just ignore their big projects that involve Reaper tech and Geth, that they appear to be capable of steering the social, political, cultural, technological evolution of humanity for the past few decades, and may in fact have the majority of major businesses (including shipyards) under their thumb, albeit almost always indirectly.
In ME3, they come out as more of a paramilitary force with, at least during the Reaper War where everyone is occupied by the Reapers, the power to enact pretty much any operation, and multiple fleets to boot. Lets just ignore (until the ending) that they actually seem to be intellectually correct about the nature of the Reapers, their capabilities, and how organics could subvert those capabilities.
What's the point? Well, TIM said it himself - CERBERUS IS HUMANITY. The Alliance may be their front face in the galaxy and the group that does most of the actual work, but Cerberus is increasingly shown to be the faction that actually pulls the strings of humanity, and has for decades since Jack Harper/TIM got into the Alliance black ops and was able to worm influence into human business, military, politics, etc. By now, there is little separating the two, and it was only in ME3 that TIM utilized his own implant controlled force because time was short and manipulating the Alliance alone wouldn't have worked during a war.
Indoctrination doesn't have to literally insert thoughts into your mind as your own. Like Kaboooom said, it reinforces the ones that Reapers want, while breaking down the ones that Reapers don't want, but they're still all of the subject's. Anything else done is just communication - like growls or dreams or whatever, all to inspire an increasing superstitious awe and nervousness about them, which creates further footholds for control. The Reapers don't typically need to directly control organics - the organics do the Reapers' work on their own.
And so it was with TIM. Even just being more evolutionary-minded at first, TIM viewed the best way to counter the Reapers was, well, to counter them, not kill them. To learn of the various truths of the galaxy, how organics and synthetics work, how organics and synthetics can be controlled. He guided humanity in that direction. We learn that the Normandy SR-1 itself wouldn't have happened without Cerberus encouraging a back-and-forth information exchange between humans and turians. We have Shepard's resurrection done by Cerberus, using significant implantation. We have AI research where TIM attempts to, essentially, create his own Reaper intelligence while it being under control of Cerberus from the start.
The actual majority or research and lab results are not really part of Shepard's soldier journey. We can just try to learn as much as we can to piece together Cerberus' progress, if we so wish. Or just reject it.
Why does Cerberus have so many ships? Well first off, its not actually that many. Its a couple fleets of relatively small size. They just have a higher standard of tech that can put them more on par with normal fleets. Cerberus basically owned humanity's shipyards (or at least enough of them) and used them to build ships under many names (front companies) until the time came to announce itself as not just a terrorist organization, but basically para-military. How? Because again, they are humanity. They do infiltrate pretty much every major human organization. Its just in ME1 that we have things more focused on it being the activities of certain individuals in the Alliance and corporations. Foreshadowing.