If your first character began as a gladiator and you've already seen the Ul'dah content, it might be worth starting your next alt as an archer, conjurer, or lancer to see the Gridanian content, or as an arcanist or marauder to see the Limsa Lominsa content. Then once you reach the envoy quest to go to other areas, switch her over to thaumaturge. (Archer you'd need to get to 15 anyway for black mage, so if you go with that you'd simply be doing your secondary class first and then your primary one.)

If you're planning on having your characters coordinate with each other for things like crafting, where you'd be passing items back and forth, you should be aware that you can't do so directly. You can only mail items to characters on your friend list, and you cannot friend your own alts as they can't be online at the same time. There are workarounds if you have a friend who can pass items to your alt for you, or if both your alts join a FC that lets you use the FC chest to pass items between them, but it's more awkward than in most games.