My opinion (not a web designer): don't let fonts define design, let them compliment it later on - I know sometimes I would be choosing fonts even before making the layout or content. For most sites, chances are web safe fonts are perfectly fine. If there is a specific and justified design vision, then choose wisely. Try and pull in one serif and one sans-serif font, at maximum. I've seen sites that slap on a bunch of fonts and quite frankly, it's not ideal. If possible, see if you can serve woff2, from a fast (or popularly cached) CDN.
