That depends on how your setup was with your old ISP. However one thing
is for sure if your old ISP hosted your domain name, DNS/rDNS records,
MX records, etc, then you should be making the request for those
information to be transferred to your new ISP (if you don't manage your
own domain name).
If you manager your own domain name, then you need to edit your domain
name record and change the name servers to point to your new ISP's DNS
servers (which they should give you) and then they will great the
additional rDNS, MX records.
Remember however, that this is best done on a weekend, as changes
require at least 48 hours to propagate across the Internet.