Because this has been a cause of such untold agony, I'll tell you this; domain transfers are an absolute headache and quite unnecessary in the first place. The process isn't exactly standardized across registrars, so every time I have to do a domain transfer for a client, it's almost always different. And it takes weeks to properly transfer a domain. Godaddy is the worst registrar I have come to deal with.
Okay rants a side. Why should you avoid this whole domain transfer hullabalu?
A domain really is like a sign-post. It doesn't matter in whose hands it is as long as you do trust that person/company aka registrar. Domain ownership changing hands is such a long bureautic unnecessary process that should be avoided at all times. It'll cause unplanned downtime making your website to be inaccessible and emails to be bounce. So don't do it.
If you want to change website or email hosts, that doesn't mean that you should also change or transfer the domain. It's just a sign post. All you have to do is simply point the nameservers to those of the new hosting company. This also sometimes -- and in most cases isn't even necessary too. Name servers are like an address book for the internet which computers use to identify where a website is hosted.
There's absolutely no technical or even business benefit of transferring a domain. Moving your domain from let's say namecheap to godaddy has no benefit whatsoever. You should always separate domain registration from website/email hosting. You can register your domain with one company and host with another. This will make it easy for you to move your website from one hosting company to another should you be unhappy with one, without touching your domain.
When registering a domain, choose a reliable registrar you can entrust with your domain for the long haul. A domain is like a title deed. It's copyright. It's your trademark. It's your business. It's very important. Guard it jealously with your life. Imagine if Facebook lost facebook.com. Then there would be no Facebook, right? It's everything. A domain is your brand. So stop messing around with it by making it change hands every now and then. Because chances are that you might even lose your domain in the transfer process.