Per LuckyBancho's last census in October, Balmung and Mateus are still the two most populated servers. But as Enkidoh said, the world visit system makes it very easy to play on other servers! The reason you don't see many people outside cities is because all the higher level people are in higher level areas or in the residential districts, and there is also the rested EXP you can gain from being in a city or sanctuary (any town/outpost with an aetheryte or residential district), so if you're going to be idle or afk it's most beneficial to do so in one of those places. Most content is played cross-world with the Duty Finder (instanced party matching) and Party Finder (recruitment board) systems so you'll be seeing players from other servers all the time. If your heart is set on living on one of those servers, you can always check the Server Status page on the Lodestone to see if a server is open to new characters: https://na.finalfantasyxiv.com/lodestone/worldstatus/

There is also the option of transferring your character if you don't want to start over, but that requires paying for that service on the Mogstation and since you only recently started you might as well make a new character.