I thought downloading Steam/installing Steam and installing the Steam client are the same thing?
That's correct Marian, the "Client"
is Steam. It installs like most other programs and you'll need to set up an account and password after activation. Once that's completed, you'll log into Steam and will have a Library tab which will contain your games. You just click on the name of the game in your library and that will kick off the game.
I loaded a screen shot of my Library tab below. You'll see the words in bold at the top of the screen; Store, Library, Community, Repeatable? Those are your tabs. The last one is your profile and displays your gamertag (mine is "Repeatable"), the name you are known by in the Steam community. That long list to the left is my Steam game Library.
My Steam Library Steam looks a lot nicer than my uploaded image. For whatever reason my screengrab was low-res.