As a developer for NameCheap.com, I won't provide my opinion on our services as that would be biased, but we charge nothing more than $8.88/yr for transfers (except for .bz, .nu domains, etc.). Features such as URL Forwarding, Nameservers, etc., are completely free and included with the registration.
You would not need to change your nameservers unless you wanted to use our Forwarding/Nameserver/etc. features. Otherwise the transfer process keeps your current nameserves in place.
"It depends on who you are currently registered with, however their support people should be able to help you through the process." Normally bad experience with transfers are due to either your current registrar not releasing the name (Network Solutions has a big problem with that, for example) or with you not having access to the e-mail address on the domain name (necessary for transfers).
Please review our website for more information, or feel free to contact me or [email protected] with more questions if you don't find the answer on the website.
Does it cost money? Unless you have a friend in the hosting biz that can do it for free, it costs money. The above posters are correct, its about 8.95 to transfer it. I recommend dotster. They transfered my domain in no time and apparently it can take from 1 to 30 days. Even when they had problems, I simply clicked the resubmit button and got the domain through. You dont "have" to have nameservers unless when you want your domain to be linked to an IP. With dotster you can add nameservers when ever you want so thats cool.
You've hopefully already finished your transfer, but I give two thumbs up to NameCheap. I just switched several domains over to them and initiated three transfers last month and it all went smooth as can be.