How to unlock the ARR beast tribe quests

In general, the next story quest in the chain can usually be picked up from the same NPC that you turned in the previous quest to immediately prior. There are a few exceptions, of course, and sometimes there are additional story quests that are picked up from other nearby NPCs (usually in the same room as the main NPC). If you do manage to lose track of where you are, though, there's a pretty simple way to do figure out what sort of quests are available to you in your current area: the Duty Recommendations list.

You can access the recommendations list through your micro menu's Duty icon; it's third from the left on the bottom row.

Selecting "Recommendations" will open the list:

In this example there are no story quests shown, but if there any quests in your current zone that you haven't picked up yet, they'll show up on this list. This list will also show you available sidequests, quest objectives, dungeons within your level range, levemetes, and other content that's appropriate for your level in your current zone. Clicking any item in the list will show you on your map where you can find that particular thing and/or open the Duty Finder window.

If you don't see any story quests in your current zone, try traveling to another zone that you've been doing stuff in lately (or zones that the story has you visiting often) and check the list to see if there's a story quest that you didn't know about. If you remember roughly where and when your last story quest took place, you could look it up in the official database, but beware of spoilers, obviously.

It's possible to make this list pop up every time you log into the game, and/or when you change zones via Character Configuration -> UI Settings -> Duty Recommendations.

According to the Final Fantasy XIV wiki:

Hunting Logs, which classes gain access to after their first class quest, are a series of one-time hunts that provide a bonus amount of experience points after defeating a certain number of enemies.

So you'll need to do your class quests in order to unlock it. It doesn't matter what level you are; it's all about starting your path down your chosen class.