Befuddled about how Steam works on new laptop Windows 8.1

[Ualdriver]

Pretty befuddled by this whole Steam thing. I bought a game TS Train Simulator 2014. It's listed as one of the games in my account where I installed in from another Windows 8.1 account/laptop. I want to install the game on my kid's new laptop (his game), so I put the disc in the drive, install the game, but instead of a normal install I get "Validating Steam Files" then "100% complete" "All files successfully validated."

OK, great. Where's the game and how do I play it?

So I figured I might have to install Steam on this new laptop, so I sign into my Steam account and I see the little "install steam" icon at the top of the browser so I click to install. I select a few selections like language, etc., but I keep getting an error message, "can't write c:\programfiles (x86)\Steam\steam.exe.

So I'm totally confused. I just want my kid to be able to play the game on his computer. Where do I even start with this?

Thanks!

[Ualdriver]

Thank you Dixie. Once I made my kid the administrator on his computer, he was able to install Steam on his computer, and then he was able to access his Train 2014 game. I appreciate your time.