Look around for a family friendly Free Company or Linkshell to join. You're going to want to be able to find groups that are understanding about it if you need to suddenly AFK for a little while, and that's more likely if you can talk to people in advance and they know that your home situation sometimes requires that. I've done FC dungeon runs where we occasionally stop for a little while because someone's kid just woke up, and everyone's fine with that, but in random groups there's a greater likelihood of the other players getting pissed off by it.

Just about any class can handle solo content, and you can spend the majority of your time solo, but as Bixby mentioned, there are also quite a few required dungeons and trials where you'll need a group. People generally recommend finding a friendly FC as a worthwhile move for any new player, but I think it's especially important to those like yourself whose family life may make your game time erratic. A good FC should be able to help you get through the required dungeons so you can advance through the game. And in return, you can help them do the same.