DNS question

[Chuck Campbell <campbell@accelinc.com>]

I've managed to completely confuse myself.

I have a domain registered at a registrar and hosted at a provider.

The provider has given me primary and secondary DNS names and ip addresses.

I have entered those at the registrar's site. All whois queries work and 
email is configured and working properly.  I can find the web site from
anyone's browser.

I now have a new company which has built commercial web pages for me, and I 
need to make them active.  This company says I need to change my DNS addresses
with my registrar to make this work.  Is this correct?  They will then take
over hosting the domain (become my NEW provider)?

They do NOT do any email, so if I make the DNS server changes at my registrar, 
will my email break?

If not, then I'm not sure I understand how any of this works.

I thought that my provider (ISP) puts up A and MX DNS records which allow
resolution of my web pages and my email addresses.  If I switch to a new 
provider that claims to not do email, who will make my email work?

thanks,
-chuck