The three gathering classes -- fisherman, botanist, and miner -- have slight variances in their mechanics, but they all work along the same lines. Once you find a harvesting node (which takes the form of a sparkling tree for botanists, a sparkling rock face for miners, and pretty much any body of water for fishermen), your menu will include an interaction for the node in question. Open the menu, select the command, and start harvesting.
First, you have to select the area that you want to harvest, with a vertical slider letting you select the exact area. The height you select does seem to have some small effect on what you can get out of the node, and you're well-advised to change height after you strike it big at one location. Following that, as a fisherman you may need to wait for a few moments to feel a nibble, while both miners and botanists may immediately start working their plot of choice.
What follows amounts to a game of hot and cold. You'll have a small framed display with a moving component. In the case of mining, it looks something like a sonar display; for botany, a pie slice; and for fishing, a sort of wave-shaped gauge. Each variant follows the same structure wherein something is moving, and the player has to press a button ("Enter" on the keyboard by default) to stop the moving component. Somewhere on the gauge is a sweet spot of resources, and you have to keep track of where you've stopped the marker each time while watching the (reasonably helpful) messages in the chat box.
There are two wrinkles that make this a bit harder. The first is that every attempt also carries with it a decreasing gauge of how many more times you can harvest -- each time you swing your axe or move your line, for instance, you get a little closer to losing sight of your target or accidentally damaging it. Once the bar runs out, you've lost it altogether. The second is that sometimes hitting the sweet spot once isn't enough, and you'll have to aim multiple attempts at roughly the same target point.
As complicated as all this sounds, what it comes down to is trying to find and then consistently hit a marked point over and over with decent reflexes. In the beginning, this will be difficult, but leveling gathering classes will substantially improve your odds of harvesting better materials. Try it a few times, and it'll almost feel like second nature.
It's also worth noting that botanists can harvest from tall grass, denoted by small sparks rising from a patch of ground. This is fairly inconsequential, however -- you just select "Harvest," and you harvest what you can from the lawn. There's no associated minigame, just freebies for having your scythe equipped.
Things get much more complicated (albeit not insurmountably) when it's time to start crafting. The process is not helped by the fact that there's no built-in list of recipes in the game, although players can use one of several online resources to find crafting recipes. (Such as here, here, here, and here.) The real challenge comes from mastering the minigame involved in making an item.
Once a player is in a crafting class, the "synthesize" option will appear on the menu no matter where he goes. Selecting this option will immediately start the crafting interface. We're going to skip the process of selecting ingredients, since it's straightforward once you know what you want to craft and made even easier through crafting quests. (Filed as local levequests, these are worth taking right from the beginning; even if you can't successfully craft the requested items, you can get skill points for making the effort, and it takes no materials from you.) The big event comes when you choose to start crafting.
The crafting interface consists of two major parts. The first is the product display, which has a large horizontal bar for progress and small numerical values for quality and durability. To the left is the menu of commands, which has four options: Standard Synthesis, Bold Synthesis, Rapid Synthesis, and Wait. There's also a small time bar that counts down after each command is executed, but you'll most likely be entering commands before it runs out anyhow, so it's not really worth worrying about.
Your objective here is simple: to reach 100% progress before the item hits zero durability. Higher quality allows you to craft +1 or better items, known to Final Fantasy XI players as HQ items. Each synthesis operation can succeed or fail, but either way it will add to progress and quality while reducing durability. Failure, naturally, will reduce durability notably without much increase to progress or quality. Wait has a special usage but reduces durability slightly, reducing it more with each consecutive use.
Standard Synthesis attempts to strike a balance between quality and progress improvements. Bold Synthesis improves quality notably but only improves progress a bit. And to complete the obvious setup, Rapid Synthesis increases progress significantly with only small improvements to quality.
All of this sounds fairly simple, but it comes out to being a fair bit more ornate in practice. Your items can be produced at a higher quality, but in attempting that you run the risk of losing the finished product each time. At higher levels, characters can choose from several options to aim for a better result or a faster one. When starting out, Rapid Synthesis is almost always the best choice, as even failures tend to keep your progress moving forward faster than your durability falls.
The other major wrinkle is the color of your synthesizing orb. At the center of your crafting attempt is a small glowing orb representing your crafted item. White orbs have the highest chance of success, no matter what action you perform. Yellow orbs are the second-best, followed by multicolored orbs having an average chance of success or failure. Red orbs, on the other hand, will almost always fail.
Wait becomes helpful here. When the orb turns red, you can avoid near-certain failure and instead wait a moment to see whether the orb changes back to a more beneficial color. By keeping a close eye on what the product is doing and the progress bar, you can produce the best possible item with the given materials. Just keep an eye on that durability!
Crafting in Final Fantasy XIV takes a bit of adjustment if you're accustomed to other systems. That being said, it also keeps you engaged even if all you want to do is craft items and sell them off at your bazaar. Once you've had a little time to get the feel of it, you can enjoy the fact that it requires more attention and care than the vast majority of modern crafting systems in MMOs.
More than anything, crafting and harvesting have moved beyond simply being sideline activities. If you really don't want to take up arms against enemies in FFXIV, you don't need to. Head from camp to camp offering your services as a freelance weaver and taking the quests available to cloth artisans. It's not quite as satisfying as sticking your sword through something, but it beats scrubbing dishes.