This can be a bit much to solely follow community recommendations, but you seem to be considering the right things. Before attempting, I'd recommend backing up your setup and noting your changes.
This isn't everything, but here are some recommendations from what I know (have broken) with Content Delivery:
If this new "my website" replaces the existing one, I would make sure you still select the same Target Type (regardless of which Publication Target you use).
Update: see Raimond's post as you don't need to unpublish the same site.
Don't forget to unpublish your existing setup. If you moved to 2013 (SP1) I would recommend decommissioning the Publication Target .
Do you use the storage layer (broker) now? If so, then yes you need to recreate or at least re-use your CD-side database(s).
You need to move or get a new license from SDL Support depending on what it's based on (typically a named server as well as CPU limits). Be sure to request or use the CD version of the license.
Be careful of files "My Site" relies on that are actually published. I've seen things like "configuration" XML accidentally checked-in to source control outside of Tridion, when they should have been re-published.
Finally, don't forget the services and file permissions you set up when you first installed Content Delivery. It's probably a good idea to follow the installation instructions again, optionally using the configuration and jars that work for you now.
Personally, I'm not sure how everyone else manages, but I seem to learn the quickest when I actually miss something.