While the amount of time spent choosing the color scheme of your website may be relatively small, it can have a great impression on the viewer. The most important thing is of course making sure that your colors do not clash, a bright green and red site will likely hurt most people eyes. I would venture that a website with badly clashing colors is far worse even than one with no color at all. However I would avoid doing this as well. The first thing I think when I log onto a black and white website is that it feels outdated and boring. Wiki sites are a possible exception to this, but their appeal is not in their design.
The Background Color of a website has the greatest possibility of clashing with any of the colors on the website. For this reason I suggest you choose a soft color either light or dark to be your background color. This sight for instance has a very soft green background. Colors can be a lot stronger than the one on this sight but if you make them to strong they are likely to clash with other colors.
The other colors on your website include everything from text to images. If any of these things are constant such as text, links and logos it is imperative that they fit your background color. Your text should be significantly darker/lighter than your background but it is not necessary for the difference to be as strong as black and white. Softer differences can often be easier on the eyes. Your logo should mesh easily with your background and the outline of the image should not take away from it.
My last point may seem contradictory but I feel it should be made. The color of your website is important but as long as it is not atrocious it is unlikely to make or break your website. In the end it is only one part of the design which is only one part of the website so do not spend all your time trying to match colors when other parts of your website are more important.