Can you copy adobe programs from one computer to another and have them still operate?

No or certainly not easily. If you're dealing with a Windows based PC you have to deal with not only the file location of the executable file that you click on (like the icon on your desktop) you also have to deal with its other file locations such as the Windows, System32, inf, etc. Not to mention the Registry, which is highly recommend to NOT be played with unless you really really really know what you're doing. No joke, you can destroy your OS if you don't know what you're doing and have to completely reinstall the OS.

This also won't work on a Mac either as Adobe tends to install files in different locations.

Now with Adobe in particular, the product key also comes into play if you're talking about paid full versions of their software. When entering that key, the software becomes "assigned" to that PC (if it's a single-use key). Adobe has a method that requires you to unregister it and properly uninstall it from the "old" PC before properly installing and re-registering on another PC.

Best bet, properly uninstall it if a key is involved and reinstall on the other PC. If it's their free software like Reader, go to their site an install a clean and updated version instead.