[Ameroth]

I could list off all the answers to your questions, but doesn't it kinda kill the newb experience when you just have a bunch of people on the internet tell you where to go, what to do, and how to do it?

But if you must:

here ya go

How do I know at what level I am able to complete Quests and Missions and how do I know wether they are soloable or not?

You won't specifically know until you try, or look up the quest/mission details online (which tend to have recommendations for level). Also, certain quests cannot even be started until a certain level requirement is passed.

What is the normal progression through levels? In other words, what are the highlights of the game, stuff everybody should do like getting a chocobo, airship pass, AF1, etc. and when can I start doing them?

There are a lot of 'highlights' in the game. Many, many different things to do at all sorts of different levels. For most beginners, some important aspects are unlocking your subjob (level 18 quest), traveling to Jeuno, and obtaining a chocobo license. Being an MMORPG, it's hard to recommend a set path for you to follow through the game. Play it how you want to.

What is the Area progression? Or, how do I know where to go next?

Move away from Windurst as mobs in Sarutabaruta begin to con weaker and weaker. You'll more than likely head out to Tahrongi Canyon, but Giddeus to the west of Windurst is a good place for lowbie levels to grind also. After you reach certain levels, people may being to ask you to go party. For the most part, levels 10-20ish are obtained in Valkurm Dunes. After that, you kinda just go to where you get invited to.