To fish, you face a body of water (possibly standing in it) with a fishing rod equipped in your Ranged slot and a lure (multi-use) or bait (single bite) and click 'Fish' or type /fish.
Based upon the rod/lure combo and chance you'll either get a "Something bit the line!/!!!" message or "You didn't catch anything."
With experience and either advice from an experienced fisherman or the Wiki, you'll be able to tell what caught your line before attempting to reel it in. More on this later.
With the camera behind you, hold the opposite direction the fish pulls your rod and you'll deplete its stamina. Deplete it fully and hit 'X' or Enter to reel it in. I personally turn the camera facing my character which makes it so you push the direction the fish pulls your rod.
You can do either, it's a matter of personal preference.
Because of sushi's ubiquitousness as the melee food of first resort, regardless of how brilliant that actually is, you'll hear people talk about how well Fishing and Cooking go together. And it's quite true. I've probably made 2/3rds of my gil in the game this way.
You can start the day with 10k gil and finish it with 250k gil from a mixture of AH sales, NPCing fish, etc. Some days are better than others though with bite rates, HQ rates, and just how fast people are buying stuff all can vary heavily.
Having at least 20ish Cooking is also useful for making your own Insect Balls to catch Moat Trout. I don't suggest completing the Lu Shang's Rod quest unless you really want to because you can sell fish and buy one a lot sooner. But Moat Carp are a decent starter skill up fish, so it's handy. More on both of these further down.
If you're serious about starting fishing you'll need to buy a fishing rod.
If you're made of gil, buy a Lu Shang's Fishing Rod. It's the second best in the game. It'll last you all the way to level 100 and lets you target the most profitable fish for skilling up.
If you'd like to start fishing, but don't have 3 million gil lying around your Mog House, then get a Halcyon Rod. It's really the next best thing to a Lu's for smaller fish. Unfortunately, while "big fish" rods do exist they're pretty uniformly terrible and I would avoid wasting my time with them.
After you've got a Fishing Rod, you're going to need bait or a lure. These are the most significant of the bait you'll run into:
Little worms: About 3 gil from NPCs, they catch small fish in "fresh" waters.For those of you who would rather not have to carry around stacks of bait that get consumed every line bite (regardless of what's on the other end with few exceptions), then you'll be mostly using these:
Lugworms: Also about 3 gil from NPCs, they catch small fish in salty water.
Insect Balls: If you watch to catch Carp you'll be using this bait.
Crayfish Balls: If you're after Nebimonites on the Ferry or Sea Serpent Grotto you'll probably use this (if you go with bait).
Sliced Bluetail: The bait of choice for catching Bhefhel Marlins on the Selbina to Mhuara (and back again) Ferry.
Meatballs: These catch the beasts of the Great Deep. If you're catcning badass fish, you're probably using Meatballs.
Minnow: Catches a wide variety of fish mostly in salt water.Fishing Skill
Sinking Minnow: Catches bottom feeding fish. Works in both fresh and salt water.
Shrimp Lure: Catches a wide variety of fish in salt water. There's some overlap and some exlusion between this and Minnows.
Fly Lure: If you're fishing up something weird in a swamp, giant tree or in the jungle you'll probably be using one of these. Also very useful in frog fishing (which is usually not very useful).
Unlike most crafts, Fishing Skill really doesn't do a whole hell of a lot. It makes it so you deplete a fish's stamina more quickly, so you're more likely to be able to reel one in.
But you'll be able to reel in fish that are 50+ levels higher than you with a Lu's (and in some cases a Halcyon). Cone Calamary, for instance, have virtually no "fight" to them, so a level 10 fisherman could reel one in despite them capping at level 48.
And there are some fish, I'm looking at you Shining Trout for my GP last week, that still are annoying to reel in even if you're 50+ levels above their cap because they have a lot of "fight" to them.
Suffice it to say manual dexterity plays a bigger role than Fishing skill in reeling in fish.
If you're borderline sometimes able to reel in a fish, sometimes not, having the added few skill points from Fishing gear can prove useful. Fishing in ToAU areas where fish are generally harder to get out of the water it may be useful.
You'll probably be fine without it though, so it's a Your Mileage May Vary sort of thing.
The only difference between the NQ and HQ is the DEF rating on them if anyone was wondering.
The Guild armor pieces are the same way except the feet give +2 skill instead of +1. The apron reduces the amount of items you'll catch which with items being generally pretty worthless means you may want to pick one up.
I'm separating these into a separate section because they're more complicated in effect and to obtain than the Fishing skill+ pieces.
The first pair are the Seagull and Heron Rings from a MMM thing. If you care about MMM, check the link and spam the correct maze for a chance at them. The two provide increase fish stamina reduction/recovery when you have Fishing Guild Support active. These look like they could be handy at lower levels, but you'll do fine without them.
Next, are the Albatross and Penguin Rings. They're rewards for completing the Pirate's Chart and Brigand's Chart quests, respectively. You can get the Rare/Ex items to start the two quests as rewards from Inside the Belly, a quest I'll be covering right after this.
Of the two the only one I'd really recommend pursuing is the Albatross Ring. That's because it's enchantment roughly doubles the amount of time a fish will stay on your line. This will make reeling in fish in ToAU areas (which are very nice for skilling up mid to high levels) much easier. For details on how to obtain it (hint: bring Summoners) read the wiki page linked to above.
Inside the Belly
Since I already mentioned it, there's a quest called Inside the Belly relating to fishing. Specifically, once you hit level 28 the NPC Zaldon at the Fishing Guild in Selbina will offer to gut certain fish for you and then give you small amounts of gil and possibly an item. The item rates are pretty terrible most or all are <10%. the Wiki page for a table of which fish may contain which items. To get Charts for the Rings you'll want Bhefhel Marlins which can be caught on the Ferry with Sliced Bluetail.
Where to skill up...
This is where I'm going to leave it for today. There's plenty of information above for a new fisherman to digest.
If you want to get started today, try using your rod of choice (release "Bad/Terrible feeling" message fish unless you sprung for Lu's) and a Little Worm in pretty much any freshwater in any of the starting cities to catch Crayfish. These cap at level 7.
Alternately, use Insect Balls to catch Moat Carp at most of the same locations. They cap at 11.
Minor Edit: I changed the sections titles to a blue I think is easier to read (and sticks out a lot more) than the bolded white text that I used elsewhere (mostly the bait/lure info).