Hi Nicholas
Moving hosting provider is pretty straightforward, personally I migrate one to the other, so take out the new hosting and run in parallel with the old hosting to make sure its a smooth transition.
Or you could just back up your site, download the files, finish with your hoster, and then take out new hosting and upload your site, but then you will have a disrupted service.
Ref domain registrant details, if you own the domain, you don't have to move the domain if you don't want to, just simply point your domain to your new hosting company. ie change the DNS records.
Or you can move your domain over to the new hosters, simply contact them and they will talk you through it.
If its a UK domain name, then UK Internet body, Nominet explains the process in full detail.
Nominet
Hope that helps.