While the rep at Moniker has been helpful and polite, he didn't even ask if this was an active site, and when he found out, informed me that they had to "cache your dns information before completing the transfer."
So, I'm pretty much on my own and in the dark. If anyone can respond or direct me to a guide on what a webmaster has to do in transferring active domains, I'd appreciate it.
Some registrars try to change those nameservers to their own namesevers, so they can get pay-per-click parking revenue from the domain names, so you'll want to make sure they don't change.
Your DNS servers might be provided by:
a. yourself
b. your web hosting company
c. your registrar
d. a third-party DNS provider
Historically, the prevailing case has tended to move from a to be to c. That is, pre-internet (ARPAnet) and during the early Internet days, people tended to provide their own DNS servers. 10 years ago, things started shifting to the web hosting company providing DNS. Today, I think the registrar is the most common choice.
You should find out just what your situation is.
If it is a, b, or d, you are good to go, as long as the acquiring registrar preserves your current DNS server information. In some cases, this means you have to be careful to choose "transfer only" vs. "parking", etc.
If it is b, it is a bit more involved. First, you have to make sure that the new registrar does offer DNS service. If they don't, you will need to make other arrangements.
If they do, you will need to make sure that they copy your DNS records (which they may or may not have access to) or else you will have to set that up yourself. You may be able to set it up on your new account before the domain is actually transferred. In this case, they WILL be changing your DNS servers to their own. You obviously do not have the option to continue to use the same servers, since your current registrar will turn off your DNS service when they transfer the domain.
Oh - to answer your subtitle question... No, you do not need to notify your web host. They are actually the one party that has absolutely nothing to do with this. (Whether they host your DNS or not.)