While the first “official” sets become available at 60, they drop in raids, that you won’t be able to solo until closer to 80. (And you aren’t likely to find groups for them as they aren’t a part of the queue system, and most people just solo them at higher levels.)
There are some unofficial sets (the ones that Nobully mentioned) that you can pick up pieces for while leveling to 60. You get the pieces from quest rewards and dungeons. (As they’re not official, they’re kind of all over the place, with different color variations.) If you clear out a bunch of quests, you should have some matching pieces. There are also some matching bind on equip sets out there. Sometimes they’ll share some looks with the quest reward gear. As I said, it’s a bit of a mess.
It’s really not worth putting a lot of time into while leveling. At max level, it will be a lot easier to collect stuff. If you just keep questing and doing some dungeons, you’ll start collecting appearances anyhow.