Contrary to some of the other posters, I actually highly recommend WP Engine since you're using WordPress. There are other similar services, but I haven't used them in a while.
In addition to the performance they offer, it's hard to beat the hassle-free nature of the platform. Caching and other stuff is handled server side. Could you do it yourself? Sure, but you're a business and time is money. The staging environment is pretty sweet for testing also. They support git as well, so version control/deployment is easy.
Regardless of who you choose, just don't use your web host for email, it's a rookie mistake. Use Google Apps or something. The only exception would be if email isn't handled by the same server.