I am a newbie owner of Wine for slackware.

All I did to install wine was download the binary package and did an installpkg on it. That seemed to make everything work fine. I can run almost all of my windows apps with it, except for those using extensive opengl screens. The app as a whole runs, but where I should see an opengl graphical output, I only see a part of the screen that doesn't get refreshed or is just blank or something. There is a thin box that is 1 pixel wide that appears around the edge of the output screen. I can see that stuff is going on in that thin line.

OpenGL works great under normal linux circumstances, but not under Wine.

Can anyone point me in the right direction?

Try using CVS. There was a bug fixed within the last week affecting OpenGL child windows.