How many email accounts are there. ?
If only a few, then it's just as easy to rectreate them at the new end, using the same passwords etc.
Once DNS has propogated, provided you use the same email address, same password and same port numbers as the ones you currently use, then your email software won't know any different and should carry on as normal.
DNS propogation can take from 30 mintes to 48 hours though, so there will be a period where your email software doesn't know which server to connect to.
Moving your emails from one server to another can be done as such.
Connect to your existing email server using the POP3 protocol and allow your email software to download the whole of your email directory structure.
Do this on an account by account basis.
When the new email server is up and running, you could connect to this via IMAP and upload the emails to the new server if you so desired.
If you are already using the POP3 protocol, there will potentially be nothing to download, so continue as you are, don't bother uploading via IMAP.
As for moving the web site.
FTP the whole of your public_html folder using something like Filezilla to your local computer then ftp these files back to your new cpanel.
Again, once DNS has propgated, nothing should appear to have changed.
Just one word of warning though, service providers like 1&1 do offer thier own web building solutions.
From my experience, these web building pieces of software rely on bits of code running inside the 1&1 infrastructure.
If your web site was built using one of these tools, then expect it not to work correctly if you migrated it away from them.
