Transferring URL

I am hosting my site through Dreamhost and want to transfer the URL from its current location to a new domain I recently purchased at GoDaddy (already migrated the new domain to Dreamhost).

I’ve tried reading forums but it is unclear to me how ‘dangerous’ or tricky this is. Can someone explain which steps I need to do and if I need to rewrite any of my libraries? Is there a way to keep both URLs active or should I forward the old one to the new one once its set up?

I need to know what to do on both the dreamhost and wordpress side.

Thanks!!!

The page I need help with: [log in to see the link]