Welcome to Eorzea!

Sadly Balmung is the most highly populated server, being the unofficial RP server (role playing), meaning it's frequently restricted to new characters being created there. However, it is not permanently locked - as the server restriction is based simply on how many people are currently logged into that server at that time, therefore the best time to aim for is during the server's 'quiet time' when most players would be in bed sleeping - a la, just after midnight through early morning to sometime after sunrise US time. Of course there are no guarantees it won't be restricted then either, but it's the best time to try.

The other way to get to Balmung is to use SE's official world transfer service which virtually instantly moves your character from one server to another, however bare in mind this is a paid service and requires an immediate one-off fee to be paid, plus there are restrictions to that as well (one being characters on trial accounts cannot have server transfers). But the up side is it seems to ignore the character creation restrictions - I myself was formerly on Ridill but transferred to Balmung late last year and had no trouble doing so. So it's definitely possible.

Either way, don't be discouraged! Keep trying to get on Balmung, as it is a very friendly server so even if you didn't have friends already waiting for you here you would soon be made to feel right at home.