It's a PHP/MySQL website hosted on Dreamhost. I've never had to scale, and don't know anything about handling traffic larger than a few people an hour.
How should I prepare for this?
Either move to a VPS Provider who can scale you up to a dedicated server if necessary (i.e. VPSLink), or go with AWS (Amazon), which basically has unlimited scale, but may also have unlimited cost (at some point, it may be better for your dedicated server to go down than to spend $60,000 in a month on server resources). Of course, this is both a personal and business choice, and really dependent on how much promise you think the site has and whether you can use your traffic stats to raise some money and grow the business.