First, manage the dns with a third party dns service. Such as http://zoneedit.com/pricing.html or similar. Don't use the dns of the registrar.
But if you are talking about moving over just dns servers (not the records that are in those dns servers) that is automatically ported over when you change registrars. But if the dns servers are the dns servers of the registrar that isn't what you want. Which is one of the reasons to use third party dns. If you switch registrars the gaining registrar automatically sets the primary and secondary dns to what it was at the old registrar.
I wrote this fast but if you have any questions let me know. This is what I do.
To answer about mail server depends on whether you mean the mail server will run on your equipment or on a VPS or?
First, manage the dns with a third party dns service.
This cannot be emphasized enough. Do not use your registrar's DNS. Do not use your registrar's hosting. Do not use your host's DNS.
The three elements -- domain registration, DNS, and web hosting -- should be segregated. That way, if you have problems with one, through chance, malice, or incompetence, you can exercise the other two items to work around the issue.
E.g.s:
If your web host is DDOSed, change the DNS entries to point to another host; minimizing downtime.
If your DNS servers are taken offline by legal action against the company that owns them, change the published servers at your registrar.
If your registrar goes bankrupt, your DNS and hosting will continue running smoothly while ICANN handles the domain management.
While you're at it, host your email somewhere different than your webhost. If you're account gets disabled because of a billing mistake, you want to get the email that tells you about it, right?
It's not even that expensive to implement these steps -- $25, maybe $50 per year. If you are at all serious about something on the Internet, spend the extra money!
You can never know what the company executives or managers might be getting up to, so -- as basic risk management -- you should prepare for catastrophic events.
I established a DNS Made Easy account about 5 years ago exactly for this reason. Registration, DNS and hosting independence is an important consideration. Separating these concerns is a big win when you need it.
But if you are talking about moving over just dns servers (not the records that are in those dns servers) that is automatically ported over when you change registrars. But if the dns servers are the dns servers of the registrar that isn't what you want. Which is one of the reasons to use third party dns. If you switch registrars the gaining registrar automatically sets the primary and secondary dns to what it was at the old registrar.
I wrote this fast but if you have any questions let me know. This is what I do.
To answer about mail server depends on whether you mean the mail server will run on your equipment or on a VPS or?