Lightroom is a whole separate application that is made for doing something different than Photoshop in general. It's, like a darkroom was back in the day. It is for developing a whole bunch of images. You can import a whole slew of photos (like, from a day's photoshoot with a high school senior). You can go through them, remove the ones you don't want, and apply effects to the ones you want to keep. You can easily apply a bunch of powerful effects to one layer or to a whole bunch at once! Basically, it's a very powerful way to develop a whole bunch of photos.
Photoshop itself (as you well know) is best used for doing very detailed work on a single image.
Photoshop Elements is just a stripped down, more consumer-friendly photo editing app.
You want regular Photoshop, but you probably also want Lightroom. Happily, you can get both Photoshop and Lightroom together in the Photography bundle for cheaper than just getting Photoshop by itself.
you need 64bit to get most Adobe products working as intended... the 32bit days are long gone mate
Which versions is most like CS3?
Well the silly answer is CS3 . CS3 came out in April 2007. Photoshop has evolved a lot in 10 years. And it keeps changing with each new release. Basic tools are the same, however.
As a paying Creative Cloud member, you can download older product versions going all the way back to Creative Suite 6 *from 2012*. That's probably the closest you'll come to Creative Suite 3. But there is no support for Hi-DPI screens, advanced 3D graphics and other major enhancements. I really think you should look at the latest version first, CC 2017, and decide for yourself.
The Photography Plan for USD $10/month (annual contract, paid monthly) is by far the best deal around. Even it you don't use everything it offers, it's still a great value.