This is really subjective, but my personal favorite is the "Creepy Crawlers" one. Find three Crawler Calculus's and trade them to Illu Bohjaa (Dhalmel Farm, Windurst Woods) for 600 gil. Crawler Calculus's are dropped by crawlers all over Vana'Diel, and are next to worthless in the stores. But three for 600 gil is quite the deal! Past levels 4 or 5, you should be able to take the crawlers nearby Windhurst pretty easily, and they drop these things fairly regularily. You could also trade silk thread in leu of calculi, but since silk thread goes for so much more in the stores, it's better to stick to calculi. You can repeat this one as much as you want.
Another great way to earn moolah early on is the "Teacher's Pet" one where you give a Two-Leaf Mandragora Bud and Bird Feathers to Moreno-Teono (Aurastery, Windurst Waters) for 250 gil. The nice thing about this one is you can repeat it as often as you want, and you can collect these items at very early levels (level 1 for the Tiny Mandragoras, and level 2ish for the birds).
Once you get to a high enough level, however, a lot of the quests become not worth the effort considering you can usually just make more money at the AH anyway.
How do I gain the ability to exit my Mog House to any part of the city?
This quest is perhaps the most useful one to accomplish very early on. Doing this can drastically cut down on time spent travelling through Windhurst. The quest is commonly called "Flower Child", although it doesn't really have an official in-game name. There is a little girl named Ojha Rhawash sitting amongst a group of flowers nearby the residential entrance in Windhurst Walls. Give her a lilac and she will tell you the secrets of the back allies and you will be able to exit your Mog House anywhere in Windhurst.
Where do you get a lilac? Well, you could wait until you can make it to Jeuno and buy one there, but chances are you'll be able to find one earlier (at an inflated price) at the AH. IMHO, it's worth it to pay the extra gil to get it as early as possible.
What's the easiest quest a newb can do without getting killed?
A decent one that doesn't even require leaving the city is "Hat in Hand" for Baren-Moren in the Hat Shop in Windhurst Waters. Get a hat and talk to as many NPCs in Windhurst Waters as possible. Don't leave the area, just keep finding people to talk to. Once you have talked to most (or all) NPCs there, return to the store and obtain your reward.
The great thing about this one is it forces you to talk to the people in Windhurst Waters, which gives you more information and quests to do and helps you learn the layout of the area. Plus, the more people you talk to, the more gil you get!
How do I do Rank 2 Mission 2?
Mission 2-2 involves going out to the canyon and killing a bunch of badguys to help the Aurastery determine the difficulty of monsters there. You must get out to the canyon and kill a lot of monsters before one Vana'Diel day passes (or, one hour of real time ;-) The mission is cummulative, so being in a party will make things easier, and it doesn't seem to matter what enemies you fight (i.e., if you just want to kill a bunch of Saplings, feel free).
Because this is a timed mission, it is advisable that you ride a chocobo out there, and have a Warp scroll to get back. The Taronghi Canyon can easily be solo'ed around level 20, but if you head in there with a group you should be fine around 12-14.
How many baddies should you kill? I really don't know. Rumor has it that you need to off 30 baddies, but I've never verified that myself (and I've heard from some friends in the game that they killed as little as 18 and everything worked fine).
How do I do Rank 2 Mission 3?
This is probably the coolest mission or quest you'll play in the game so far (before you get it). This mission involves killing a Dragon!
The gist of it is that you must go to each of the other kingdoms and perform some sub-mission there. Depending upon which kingdom you go to first, you will have different quests. I would recommend doing this quest sometime after level 20, and doing the second half of the quest (the Dragon part) with a party. The Dragon has a level cap of 25, so if you are higher than 25 be sure you have armor that is 25 or lower. The Dragon really isn't too bad for a group of fighters level 20-25.