How do you transfer a domain name without interrupting service?
A transfer of registrar doesn't change the DNS servers registered for the domain.
If your old registrar was also your DNS hosting provider (and if that DNS hosting service was contingent upon your registration) you should probably get new DNS worked out before you perform the transfer, to be safe.
It sounds like what you're really doing here is CHANGING REGISTRARS.
If that is the case, most registrars automatically adopt the DNS settings of the old registrar and the switch is seamless.
BUT... if you are also HOSTING with your registrar you'll want to make sure you have a copy of your entire website moved over to GoDaddy ahead of time. Otherwise the switch will take place and you'll be pointing at nothing more than a parked page.
So, the question is... where are you hosting and who is providing your DNS right now?
If you are not doing DNS or hosting with your old registrar you should be just fine.
One last note... if this is your "first rodeo" its usually nice to do this sort of thing on a Friday... just in case you screw up you'll have the weekend to try and fix it.
DNS propagation can be somewhat unforgiving sometimes. :-)
Good luck!