I've finished the noob-friendly walkthrough so that people can get a sense of what they do with the interface, and how it would work for them. Hopefully, more people will be able to look at the interface, and comment on how it works with that guide.
"But what do I actually do as a Player?"
So, rather than an overview of what each button does, let's instead go over what you would want to do when you are first setting up a fort.
"I just embarked, I want to farm plump helmets, since those are traditionally fast and easy to farm."
OK, so dig yourself some nice open space in your new fortress for your farm to actually occupy. Once that's done, you want to open up your land management menu, and get to designating the actual farming zone.
You'd open the menu with "l" and need to start designating the farm with "w" to bring up the second tab, the farm designation window. From here, zone out your farming plot. There's no real need for "building" a farming plot in this system, it's just designated as the land that your dwarves will work when they are trying to grow something.
To actually get them working, open up the scheduler window by hitting "e" to activate that tab. We will assume you actually brought plump helmet spawn, and haven't eaten them all or something silly at this point. So, what you need to do is create a new farming block, for a good three or four months, and then scroll over, and select the plump helmet seeds. In all odds, if you just dug out the place for this farm plot, you will need to add to the biomass and water levels of this plot to make it suitable to the plump helmets' growth. Plump helmets can survive with little existing nutrients and even soil of any kind, but grow on rotting material like logs, so you can simply add several logs to the farm plot, and keep them wet to both grow plump helmets, and also build up the soil nutrients, paving the way for future underground crops to be planted in the same area. To do this, you need to add some permissables - use "z" to start adding permissables of providing water by bucket and logs to grow those plump helmets on. Use the + key to keep adding the number of logs you will permit the dwarves to apply to the ground until the ♠ symbol turns green. Make sure the ≈ symbol is green as well by adding to the number of permissable buckets of water to apply to the soil, and use the * key to add ten bucketfuls at a time, or - until you hit a negative number, which should make an ∞ symbol appear, designating infinite uses of that (presumably plentiful) resource.
This means you have set up a schedule for that farm to be used for plump helmet farming for several months, and have given permission for the dwarves to use the resources they need to run that farm. Dwarves will determine whether they need all the materials they are permitted to use or not at the time, but permissables give you a chance to throttle how much they are allowed to use if you have some other priorities for those resources. The logs you are using, for example, are potentially useful in other industry, and you wouldn't want to throw all your logs at just farming if there are other uses you have for those logs.
Speaking of which, you probably don't have enough logs to sustain that farm forever. You need to designate some area for clear-cutting to generate the wood you need for your farms and future industry. The land management menu has replaced the old designation menu for the purposes of designating trees for cutting down. Now, you designate a zone where trees grow (similar to the plot designation), and then set the area to be continually chopped down for lumber. You do this with the "a" Forestry menu. You can also set it for herbalism at the same time to gather some easy food to tide you over until your farms are working at full capacity, and get you the seeds for aboveground farming. Herbalism, in fact, is a skill with a simultaniously more defined and less defined role now that farms and wild soil are made to be fairly similar. Herbalism easily supports small numbers of dwarves that can manage to keep their harvesting areas safe from skulking goblin ambushers, but at the same time, plundering too many plants from the soil without returning nutrients can gradually lead to a more barren landscape with less plants overall.
Also, while you don't need to flood your fields anymore, if you have bucket brigades watering your fields, then the closer you can get a replinishable basin of water to your farms, the less distance the dwarves have to ferry water by bucket from the basin to the fields. Maybe later, you can improve this system by building irrigation pipes connected to a pump or sluice that will take the "bucket brigade" out of the picture entirely.
Now let's assume you're ready for aboveground farming with some of those seeds you gathered. You've walled in an area, and also managed to cut down or gather all the trees and shrubs in the area, and need to have a denser, more sustainable food source. Now you can start really building your first self-sustainable farm system, one which doesn't require outside resources aside from the manure the living creatures in your fortress generates. To do this, zone another farm in this aboveground area.
Why do you need to go through this trouble? Well, plump helmets require a constant supply of more trees cut down to serve as their biomass. This means that eventually, you're going to be needing hundreds of trees cut down each year to just keep your plump helmet farms going, not even counting the rest of your industry. That can get unsustainable fast.
Now, you're going to need to schedule a more complex system than you did before with the plump helmets. First, make a block to harvest some staple crop in. Let's assume it's wheat, and wheat takes 8 months to grow, starting from Slate/mid-Spring to Timber/late-Fall. Wheat is a "heavy feeder", and needs plenty of nutrients in the soil to grow. We'll assume that the soil just happened to already be mostly ready for the purposes of this demonstration, but you might need to spread several layers of fertilizer or grow some preliminary crop to prepare the soil for this wheat. Growing wheat will probably require permissables set throughout the year for several applications of manure, water, and it will probably be a good idea to give permission to till the soil before the wheat is planted (this kills weeds).
Now, because wheat is a heavy feeder, it will severely deplete the soil's nutrients, so we need to have a system in place that lets soil fertility climb back up, and it needs to do this without claiming hundreds of trees per year that you may not have on hand to keep feeding your agriculture industry. This means crop rotations. You can first try to at least maximize your growing season, however, and try to grow something small and light-feeding, like onions, as a winter crop. Light feeders won't consume as many of your resources, and may grow faster than some heavy feeders, but generally aren't as economically valuable or capable of feeding as many people as your heavy feeding staple crops usually will. They will, however, give you a chance to really use up the last of your soil fertility while you go ahead and start adding more manure and biomass to allow soil bacteria a chance to break down your fertlizers into usable nutrients while still giving you something of economic value from your fields.
Now you need to schedule what you do for the next year, and we want this year to be based around the idea of replinishing the soil for the wheat the next year. This means it's going to be a two-year crop rotation cycle, which has the advantage of simplicity, although it isn't as efficient as more advanced crop rotation cycles.
Instead of merely "fallowing" the field, which lets weeds take over, we will instead schedule clover, which can serve as fodder for livestock. Clover is a soil replinisher, it actually adds some nutrients to the soil as it grows, although it still does take up some other nutrients. During the time that clover grows, however, we can send in our cattle and livestock as grazing herds, and let those cattle leave their own fresh fertilizer behind on the fields as they eat - giving us back some of those nutrients that the clover wasn't replinishing all by itself. We can grow clover for part of this "set-aside year", and another replinishing crop that produces an edible crop, like peanuts or soy beans, during their particular season. During all of this time, we keep on adding manure, helping push up the soil fertility in anticipation of the heavy feeders to come next year.
At the end of this second year, we can then set the scheduler to be on "repeat". The wheat will then be slated to be grown the next year, and before it is planted, the soil will be tilled, fertilizers will be added, the wheat will be sewn, and wheat will be grown, watered, and harvested, all on auto-pilot by the dwarves. A single scheduling session at the start of the first year is all you need to do to set this up.
Of course, that doesn't mean that farms will run perfectly forever - pests may strike, fertilizers may run low, droughts may occur, but generally, you can leave the day-to-day operation of the farm in the hands of the farmer dwarves, while you can focus on other aspects of the game.