What methods are available to keep a Windows program looking the same despite the resolution they are run at?

I'm writing a program on my computer with 1600x1200 resolution that, when run at a lower resolution, looks terrible. The button size and placement is off, the text looks anemic, and on one computer (that's actually at 1600x1200) some of my bitmaps are being clipped by the bottom of the window.

How do you do it? Any suggestions?

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Rutabega