I figure this would probably be a good place to see if anyone is knowledgable on the subject.

So I've come across a surprisingly excellent maintained copy of the X-Wing Collector's Series on CD-ROM in a hand-me-down store and of course, Windows 7 64-bit is being fussy. Booting up the first disc brought me to the little pop-up of the usual "Install X-Wing/Install Flight School/Check Your Computer/View Documentation/Exit" things. Clicking on the "Check Your Computer" button brings up the little screen showing my computer's processor speed and so on. It's fine, I meet all the requirements but it says I'm running on Windows 95.
So I go to install it and 'Oh no!', it wouldn't let me install it because it could not find the disc in the disc drive. Any tips?

Try running the installer as administrator? Sometimes elevating these kind of things alleviates weird errors.

Better yet, you might be better off just making an ISO, finding a no-CD, and mounting it with PowerISO or something similar.

Did you try running it in 95 compatibility mode?

1) The drive with the disc should have the first letter possible (for example, your harddrive is C, the driver with the disc D and another driver E). Ingame music wont work in X-Wing and TIE-Fight if this isn't the case.

2) The old installers don't work on 64 bit OS. Check out these fine installers: http://www.markusegger.at/Software/Games.aspx

This. Installshield for Win95 was a 16 bit application, not supported by Win 7
The site above will provide zip files that contain newer installer versions. The trick consists into copying the content of your original cds to a temporary folder, replacin the old installer by the win 7 compatible one and burning new install cds. It works ( although even with that , Tie fighter may present some graphical bugs ).