Who is your domain with ?
1: The usual process is you get hosting with your new provider
2: Set up your site there ...
3: initiate a domain transfer (through blacknight if you're using them)
4: respond to the verification email that the domain holder receives)
5: Other crowd will go ok ... and it'll go over to blacknight
DNS propagation could take up to 48 hours ... but so long as you've got it in both places you should be fine.
Email is another matter but you've not mentioned that.