Google might not support direct data migration/transfer between custom domain (yourname.com) and gmail.com. Of course one can use Google Takeout to download from one, but it might not be easy to upload/import back into second.
But, for your stated requirements, you don't need migration, as you are ok with two email accounts.You already have email@example.com. What you essentially need to do is purchase mysite.com, add it to Google Apps (not free as of now), and then you are free to create any email there, so make firstname.lastname@example.org
You will end up with two emails, both having gmail backends, @gmail will be standard, @mysite will be Google Apps. Later either you could auto forward (& send)emails from one to other, or delegate access, or use multiple logins in Chrome.
Delegate access might not work between different domains, but you can always connect one account in second, in gmail settings, and then can send and receive emails from both in one place.
Calenders can be shared between both accounts with variable access.
Google Drive - Google always keeps the originator account as owner of files/folders. The workaround I have found is share a folder between both with full access, and all child files/folders will carry parent folder's permissions.