Yo, I think I have the answer you are looking for. That answer, to put it bluntly p, is no you cannot move PS3 games of any form to the PS4. The PS4 isn't backwards compatable at all (though I wish it was).
What you were hearing about was Sony's Playstation Now service, which is powered by the Gaikai service I believe. It currently allows Ps4 and ps3 Systems to stream select games to their system of choice. It's library is currently limited, but they said they plan on expanding that library greatly (and I mean GREATLY). They also plan on bringing that service to PS Vita (including the vita tv) some sony brand TVs, smartphones and tables, and lastly PC (I believe that's all of them).
Right now the only option for Playstation Now is to rent single games at the prices they list for each games with different prices for different amounts of time. Sony says they plan on making a subscription plan, but they released very little details about said plan (as far as I'm aware). So the best option for you right now is to just keep the ps3 until either: A) Sony releases many more details about that PS Now subscription plan. Or B) there are enough games released on the ps4 to warrant owning both consoles, provided you have the money to do so.
I hope I was able to help you in any way.