First, create the logo image at the highest resolution you can deal with. Now, this first assumes that you have read up on what exactly "resolution" means: viewtopic.php?f=34&t=32569 . For instance, changing the "Resolution" from 96 DPI to 300 DPI does not change the fidelity of the image as it appears on the screen. It only affects the default physical size used for printing.
For instance, the Paint.NET icon (the clouds + mountains + paintbrush) was authored at a pixel size of 2048 x 2048. It is then feasible for me to reduce it to pixel sizes of 256x, 48x, 32, and 16x, which are all the standard Windows icon sizes. If someone asks me for a "print version", that usually means they want 300 dpi. So I set the image to "300 dpi", and send it to them. In reality, the only thing that happens when I set it to "300 dpi" is that the number 300 gets written into the file instead of, say, 72 or 96. The pixels do not change, and the image has not actually "grown" or "shrunk". It is only a hint for your desktop publishing or word processing software with respect to how large you intend it to appear when printed on physical paper.
Also, you should always use .PDN or .PNG. If you use .JPG or .GIF, they will mangle the quality of your work. Those file formats are intended to be used for photographs and low-bandwidth web images, respectively. Not logos.