If you already own a domain (gilo05.com), then all you need to do is:
1: Sign up for O365 and setup the tenant
2: Verify that you own the domain - this is done in O365
3: When ready to switch, modify your MX records (in your DNS) to point to Microsoft. The domain setup wizard in O365 will guide you through this and may even do it for you if you have a partnered DNS server. The key records are the MX (mail exchange) and SPF (TEXT). However, setting up DKIM is recommended as well.
Changing the MX record is like going to the post office and filling out a change or address form. The MX record tells all other mail servers where to send your mail. Changing MX will make all mail go to Microsoft. This usually starts almost immediately but can take up to 48 hours globally.
Now, the one topic not addresses is mail migration. If you have mail setup somewhere else, clearly there is a collection of data there. You may opt to simply leave it behind. Or, you can drag and drop it between the two accounts. Or you can use Microsoft's migration tool.
You can keep your domain at the current registrar. This is all about DNS changes.