I would suggest that you transfer your domains to a domain registar, not your next host. That way you dont have to keep paying the transfer costs for the domains. So pick a good registar, such as godaddy.com
Once you have your domains transferred to a registar then choose your new host. You may transfer your domains to your new host, but if you dont like them in time, you will once again have to move the domains. So i would suggest a stable icann registar. Then find a place to host your web site.
Totally. Go with someone like GoDaddy where you can better control your domain. With so many host falling under these days, you could potentially loose your domain or should I say, access to it. Keep it in your control.
I have never heard of NameCheap, I personally put all my business and client domains through godaddy at this time. They are very reliable, have good support, and will always be there.
It up to you who you choose, but i would look around here for reviews on anyone before you change.
Yes, move domain to the new registar, takes up to 72Hours saying your previous registar dosen't drag their feet. While they are working on transferring your domain, I would start looking for a new web host, look here for reviews. Then once that is all set your domain should be ready to set the new nameservers.
I would highly recommend NameCheap for this, i use NameCheap for most my personal domains, and some of my clients, free push to other NC users, simply the easiest management tools and a real friendly support team
1) Backup your site
2) Tranfer domain (to Godaddy or namecheap) and update nameservers
3) Search for a good host
4) Transfer your site to the new host and update nameservers
5) Grab a pizza and coke and watch the latest show on tv