What is the best way to migrate multiple SQL Server databases?
Copy 50 to 100 databases to a completely new server with zero downtime?  I don't think that's going to happen.  When you're talking about a migration, planned downtime is an absolute must.

In this case, depending on the size of the databases, I'd take the scripting route with this.  If you are proficient with PowerShell, you can loop through all the databases and take backups to store in accessible storage for the new server.  Then on the new server, you can script out the restore of all the databases.

Don't forget about the server-level items (logins, jobs, etc.).  You can script these as well, but with this particular requirement SSIS actually makes it very easy.  There are tasks to do all of these server-level items.

If you're expecting no downtime, how are the clients going to redirect their requests to the new database server?  There are a few external solutions for this, but something to consider when you promise "zero downtime".