The PS3 and PS4 have dramatically different architectures (so do the xbox 360 and xbox uno), making backwards compatibility functionally impossible. Don't get rid of of your PS3, that's for sure.
I second Leoofmoon's suggestion to wait on getting a new system. At least wait until we find out at E3 if Sony will release a new PS4 SKU with a larger HDD installed. Then you can decide whether to get an older PS4 (which should drop in price) and keep its drive or upgrade it yourself. (It's an easy task for both PS3 and PS4. I put a 120 GB drive in my 40GB model PS3 years ago.)
A major complaint about this generation is both PS4 and xbone have 500 GB hard drives, with maybe just over 300 GB of space open for mandatory game installs (disc or full download, both require large files being on the HDD). The mandatory installs are usually between 30 and 50 GB, so you'll get around 10 games (minus space for patches, DLC and PSN titles) before you have to delete old data to make room for another game. And, if you wish to replay an old game you deleted, you are gonna have to wait for it to reinstall again. It's best to future proof yourself with a larger HDD right off the bat.
Another thing to watch for is that Playstation Now service, the one that let's you stream some PS3 games and PSone Classics to PS4s, PS3s and a few other devices, all from Sony's servers. It's not the best option, since it requires a solid high speed internet connection, probably still has noticeable input delay, and seems overpriced. (Personally, I don't like the concept, but I might as well mention it. It seems to me like a dark sign of a future streaming only generation and "games as a service.")