Web page hosting and Email hosting

I have a long-time client who I advise for all his computer office needs –Windows, hardware, office software, etc. But when it comes to web/email hosting, I have only basic knowledge, and not enough of that.

So here’s the situation: this client runs a small business, and has used a local web hosting service to host a business web page, and a different service to host host his email. He has his own domain, so his email address’s domain is his business name.
Well, these hosting services are fizzling out — he’s tried repeated to contact them with no luck. Lately, he tells me his business email is no longer working, and attempts to have it rectified seem to fall on deaf ears.

So, my question is this: how do I help him move to new, reliable hosting services? Who does he sign up with? And how does the new hosting service “take over” his web page (I assume that’s a matter of copying the HTML code and pasting it into a new hosting service, then re-registering the domain)? If he’s using IMAP email, how is that content moved to a new server? It’s not likely we’re going to get much in the way of cooperation from his existing services if they won’t respond to his pleas for help.