Optimizing a website requires choosing fonts that are readable, accessible, and also with relatively fast load times. I prefer to use open source fonts and Google Fonts is my go-to. Google also provides useful usage information, such as geography and number of websites that feature the font.
I usually use sans serif and serif fonts for the main parts of the web page. Here is a list of fonts I recently used for websites I worked on.
However, if I do need to use display or handwriting fonts, such as for logos, I try to create an image instead of loading the fonts. I want to make sure that the page looks the same across browsers and some of these fancier fonts can give you inconsistencies.