Christian (just for MY knowledge), what is the reason of moving from Windows Server to Linux?
Now for your query, in moodle you'll find an experimental script, Administration Block > Site Administration > Development > Experimental > Database migration.
As you'll be moving OUT from SQL-Server so there must be MS-SQL drivers installed in your host machine, you have to enable PHP drivers/extensions for mysql/mysqli drivers as well so you'll have all database drivers/extensions for PHP available in your machine (you can as well install MySQL/MariaDB on same machine as they make use of different ports as MS-SQL server).
The tool I mentioned above will show you option to choose from target DB to convert to, can be local or remote machine, like showed in image below.
In above image, my Moodle resides on Moodle server on MS-SQL Express Database, and all I had to do in enable extensions for MySQL to give me options to choose from different DBs.
Just a word of warning, this is highly experimental tool, so may not work 100%, just keep backups of DB or ideally use another machine to import MS-SQL and convert on same machine - thoroughly test and then migrate.
Now as you mentioned being a newbie to Moodle, this above method MAY be a bit overkill for you, alternatively you can export courses with users intact from MS-SQL moodle, do a fresh install of DB and Moodle on Linux/mySQL and import courses way, that would be a certainty that all DB has clean code.