Yes, there is open source and free software in every category I have ever needed, I only very rarely need to go to paid and licensed software for features or performance. It is easy to mount an argument that only open source software can be secure. Paid software is rarely open source.
As to which software you need depends upon just what you want to do.
Nearly everyone who puts a website onto the internet pays someone to host that site, the operating company looks after all the 24hr power and connection issues, maintaining computers, updating server software, defending against attacks. There are many companies doing this very well and very economically, in terms of business and even hobby expenses it is a very modest expense, even cheap, presumably because it is also very competitive.
If this is what you want to do, then you can use Filezilla, you will need it as a client to manipulate the files up on the server you hire.
Of course as website operator you are also obliged to assist in maintaining a secure website.
If you want to do your website development on a localhost server and then transfer your developed sites and components up to hired hosting, then you would use an FTP client like Filezilla for that too. Doing development in this type of environment can be very beneficial and productive.
If on the other hand you want to use one of your own computers to serve your website(s) out to the internet then I think you are very brave.
If in addition you want (either yourself, nominated people or the wider internet) to be able to use a client (like Filezilla) to upload/download files your server, then your server would need an FTP server. Note that it is only in accommodating connections from FTP clients that would cause you to need an FTP server. For instance it would be possible to login to your WordPress website on your server from out on the internet, and then use the WordPress admin capabilities to for instance upload images or download files your website was configured to deliver.
It goes without saying that using your own server, while it makes you lord of the realm, it also makes you totally responsible for updates and security.
Hope this helps.