Where can one download the office bitcoin core software? The bitcoin software that everyone is using. Just typing in "download bitcoin" in google, its quite confusing for newbies as you have bitcoin.org and then bitcoincore.org - which one is the reliable copy? Is there another place I should get it from? And who even controls bitcoincore.org?
Those are both legitimate sources. You can be confident in your download by verifying the checksum and the PGP signatures provided on the respective sites. The process is outlined in more detail on their download pages.
I think the method with the highest level of confidence would be to clone the official repository: https://github.com/bitcoin/bitcoin and build it from source.
I don't know who controls bitcoincore.org, I'd assume the maintainers of the Bitcoin Core developer community. They are linked in the GitHub repository, so thats a reassuring signal to me.