For when you launch the new website, Google have a change of address tool:
https://support.google.com/webmasters/answer/83106?hl=en
As mentioned, 301 redirecting the exact old page to the exact same new page is the most important step. Plan ahead and map out your 301 redirects. You want to be sure of what's happening when the domain is transferred to give your rankings and traffic the best chance of remaining intact (or even increasing, which I've seen with some website redesigns).
I think you can change your Twitter handle more than once. If you have a good existing following I'd recommend updating your existing profile to the new handle, and (if possible) re-registering the old Twitter handle at another account - linking in the bio to your new Twitter account. That'll make sure you retain some of the link authority your current Twitter profile has built up and ensure people can still find you through any old links. I think custom Facebook URLs can be changed once.
I actually wrote a site migration guide last week, there should be a few points in there you'll find useful: insideonline.co.uk/website-migration-guide/