Selecting fonts for your website should never be rushed; it should complement the colours, design, and space on your page.
Making sure the copy is practical is probably the most important first step, and a straightforward font is important when it comes to the body copy of your articles. There is room to be creative and have fonts which take a similar appearance to a company logo, but it is important to use these carefully and not to overdo it.
Well designed and thought out websites tend to have a mixture of fonts for different purposes, and choosing the colour of the text is just as important as the style itself. Again practicality comes into it, as the text should be easy to distinguish itself from the background, reds on to of black should usually be avoided, and vice versa.
Typeface sizes can provide another headache, and again this comes down to practicality, the bolder and larger the text is, the more eye catching it is, and if there are pieces of unimportant information underlined, bolded, or made too large, it damages a professional look.
Changing fonts throughout a website can have a similar effect as the design itself, people do notice, so think hard before you commit to a font, there may be a lot more riding on it than you think.