Moving our Website

A company that I consult to hosts there own web site. We are considering moving it to a company that would host it for us. We have a domain, for this discussion I will call it company.com. Will I have any problems if I move www.company.com to a hosted site? We also have our own email client. Can I keep our email on our server and move the web pages to another hosted server? Is there anything that I need to know to make all of this happen smoothly?