Okay, I'm sure that someone else (if not everyone else) has managed to find this solution before me, but I'm posting here what I've discovered through trial and error.
If you installed Redguard to the default directory, go to "c:/GOG Games/Redguard/" and look for something called "DOSBox Configuration". It's a shortcut, from what I can tell. Double-click it. A small window will pop up. On the Basic Settings tab, change the graphic mode to Direct3D, and check the "Full Screen" box. On the "Advanced Settings" tab, set the window resolution to your preferred resolution. I recommend whatever you have your desktop set to. Save your settings, then click Start Game.
If this doesn't work for you, tweak the settings and try again. On my PC, it still doean't run buttery-smooth, but it's actually very playable, as opposed to what it was set to "out of the box".