There are quite a few methods you can use to do it.
There are other methods like using the Call Operator (&), Invoke-Expression cmdlet etc. But they are considered unsafe. Microsoft recommends using Start-Process.
Method 1
A simple example
Start-Process -NoNewWindow -FilePath "C:\wamp64\bin\mysql\mysql5.7.19\bin\mysql" -ArgumentList "-u root","-proot","-h localhost"
In your case
Start-Process -NoNewWindow -FilePath "C:\Program Files\IIS\Microsoft Web Deploy\msdeploy.exe" -ArgumentList "-verb:sync","-source:dbfullsql=`"Data Source=mysource;Integrated Security=false;User ID=sa;Pwd=sapass!;Database=mydb;`"","-dest:dbfullsql=`"Data Source=.\mydestsource;Integrated Security=false;User ID=sa;Pwd=sapass!;Database=mydb;`"","computername=10.10.10.10","username=administrator","password=adminpass"
In this method you separate each and every parameter in the ArgumentList using commas.
Method 2
Simple Example
Start-Process -NoNewWindow -FilePath "C:\wamp64\bin\mysql\mysql5.7.19\bin\mysql" -ArgumentList "-u root -proot -h localhost"
In your case
Start-Process -NoNewWindow -FilePath "C:\Program Files\IIS\Microsoft Web Deploy\msdeploy.exe" -ArgumentList "-verb:sync -source:dbfullsql=`"Data Source=mysource;Integrated Security=false;User ID=sa;Pwd=sapass!;Database=mydb;`" -dest:dbfullsql=`"Data Source=.\mydestsource;Integrated Security=false;User ID=sa;Pwd=sapass!;Database=mydb;`",computername=10.10.10.10,username=administrator,password=adminpass"
This method is easier as it allows to type your parameters in one go.
Note that in powershell to represent the quotation mark ( " ) in a string you should insert the grave accent ( ` ) (This is the key above the Tab key in the US keyboard).
-NoNewWindow parameter is used to display the new process in the current console window. By default Windows PowerShell opens a new window.
References : Powershell/Scripting/Start-Process