I recently bought a copy of RCT2 from GOG to install on Windows 8.1, and the installation worked fine, but when I fired up the game, I noticed that the program was struggling to reach a greater speed. I checked the support for this page and tried everything I could to fix this, but the overall result was the same.

Out of frustration, I decided to let it slide and play it anyway. Then, when I went to rename a ride, the speed of the game picked up to a faster, more desirable rate, only to slow down again when I finished renaming the ride. The same thing happened again, when it started to rain in the park, only to slow down again when it stopped.

Is there a certain graphics setting that's causing this, or has this happened on other copies that were bought on GOG?

Have you already tried running the game in compatibility mode and with admin rights (right-click the game's shortcut, select Properties and look at the Compatibility tab to set these options)?

I have already done both before and after this was suggested, but the results were still the same.

I noticed the same issue a few years ago. IIRC you can open one of those rename windows and relocate the window into e.g. the very right corner so as to almost hide it and then you can play at normal speed.