I've used the watercolor effect on one of my logos. When I try to open the file using a program other than Inkscape (such as Corel Draw), the watercolor effect is just shown as a circle. What format should I use to save the file so the watercolor effect can be seen?

Do you mean you applied an SVG filter from Filters menu? Corel DRAW probably doesn't support those. You cannot do anything about it other than exporting to PNG and edit it as a bitmap.

'Watercolor' is implemented as SVG filter effect. If you save a copy of your file as PDF (with default export options), any objects/regions with a filter effect will be rasterized and inserted as bitmap into the PDF file, thus keeping the same look but no longer being editable as vector objects.

Note: I don't know if current versions of CorelDraw support SVG filters only partially or not at all, nor do I know if or how well it opens or import PDF files.

[Edit] If CorelDraw supports linked or embedded bitmap images in SVG files, you could alternatively create a bitmap copy of the filtered object in Inkscape ('Edit > Make a bitmap copy', and then embed it with 'Extensions > Images > Embed Images…') and hide the vector object for further editing (in Inkscape) on a separate layer. [/edit]