How to level up?

[Anova]

Low-level roulette is actually unlocked starting at Lv.17 or so (assumes you can run Sastasha, Tamtara, and Copperbell).

At Lv.10, you should be able to do guildhests. Those will net you the largest chunk of xp solo, as they will award you with the first completion xp bonus for each class you clear them with. The Guildhest roulette bonus xp will also help on a daily basis

Barring that, completing the quest, Rising to the Challenge, will unlock the challenge log and help augment your xp gain on a weekly basis.

Your Lv.10-19 hunting log should help you reach Lv.13 or 14, although you may have trouble killing all the monsters. The guildhests should help with at least 1 level as well. Consider running some FATEs in Western La Noscea if you fall short.

Each starting town has 2 unique quest lines (approx. 10-12 quests) only available to players who started in those towns. Since you started in Limsa Lominsa, you do not have access to their stories. It's sad to say, but you cannot depend on quest xp when leveling alt classes. Instead, you gain a 50% xp bonus on FATE xp and killing enemies for your alts. For that reason, the hunting logs, FATEs, and roulettes are your best bet for leveling alts.

[Niwashi]

Low-level roulette is actually unlocked starting at Lv.17 or so (assumes you can run Sastasha, Tamtara, and Copperbell).

You need at least two dungeons to unlock Roulette, so if all of those starter dungeons were already unlocked by a previous class, it would be at level 16 when both Sastasha and Tamtara are available that a new class could run Roulette.

However, it sounds as though the OP stopped in the main storyline once reaching Ul'dah, so that needs to be advanced a bit as well in order to unlock those dungeons. If you plan on making THM your new main class and doing the main storyline with that, you'll need to advance to level 15 and then pick up the next storyline quest, which will unlock the Sastasha dungeon. From there on, the main story and dungeons will be the way to level.

(The next part of the main storyline will be dungeon-focused, where you have a quest to complete a dungeon, then a quick quest to report to the NPCs how it went and be told where you're needed next, then a quest to run another dungeon, then repeat for a third dungeon. The dungeons have to be run once each to progress through the story, but can then be run again for additional XP and loot.)

So the only real issue is with getting the rest of the way from 12 to 15. I had the additional advantage of a couple hats from Pre-order and Collector's Edition bonuses that gave an XP boost through the first 10 levels, so I was getting nearly to 15 by the time I completed the hunting logs and class quests, and just needed to pick up a little extra from a couple sidequests. (There are some sidequests available in each of the regions, although not as many as there are in the region where you start the game.) Without those boosts, it might require a little extra grinding with a combination of FATEs, guildhests or guildleves, but 15 isn't all that far to get.

p.s. The challenge log some people have mentioned will help eventually, but I'm not sure how many of its challenges are available in the 12-15 level range. Many of its challenges involve doing things you haven't unlocked yet.