The only thing I can speculate is you are also using cooking (which you must be to be eating turkey eggs), and you haven't turned off cooking plump helmets and booze etc in the status->kitchen menu (you can't turn off cooking until you actually have the item in your stocks, so you'll need to revisit the kitchen page several times), this will result in the plants being cooked, destroying the seeds, or result in the booze being cooked, destroying the booze. You can't turn off the eating of plump helmets etc, but dwarves will normally go for the nearest food they can find, so putting your plant stockpile out of the way, further than the prepared food stockpile, will cut down on plant eating.
Once you've dealt with the cooking situation, basically here is how farming works:
1) You don't need many farm plots. A few 3x3's is plenty.
2) Make sure to set planting orders for every season (a)(b)(c)(d).
3) You should only use a couple of farmers. More highly skilled farmers generate much larger stack sizes. But turn off all other labours on these farmers (perhaps allow them food hauling), also under the orders menu, set it to "only farmers harvest" so that the farmers gain the planter experience from harvesting.
4) The general consensus is that fertilizing is not worth it. It's resource intensive, and your better off with more skilled farmers and more/larger fields.
For brewing:
1) Make sure you have lots of spare barrels, or large pots (green glass or stone work well, not all pottery pots can hold booze particularly unglazed ones made from low-grade clay), embarking with a stack of logs (or bit. coal and sand bags, or stones) and making lots of barrels (or glass pots, or stone pots) is a fair idea.
2) Brewing doesn't have a quality level, so you can use as many brewers as you like, but use multiple breweries if going for low-skilled brewers.
3) Remember that dye plants (dimple cup, hide root) etc cannot be brewed. You might want to disable these plants from your plant stockpile.
Other:
1) Herbalism is a good investment for a starting dwarf on most biomes, the points give a very good return in terms of plant stack size (a proficient herbalist will sometimes bring in stacks of 4 or even 5), I'm an experienced player and I like Herbalism because it's good for quickly bootstrapping surface plant farming, and for laziness. It's even better if you're new and totally mismanage things, if you need some plants, just harvest some. It doesn't work nearly as well without points in herbalism, but is better than being sober.
2) Caravans will bring you a lot of plants if you don't have any (if you turn your plants into something else, like booze, the caravan still thinks you have a plant shortage), dwarves bring plump helmets, elves and humans bring surface plants. You should learn the ins and outs of farming, but between herbalism and caravans you can produce a great deal of booze without farming. It's often a good idea to buy out the caravan's plants, because the stack sizes are larger than you get out of farming (sans fertilizing), this means bigger pots of booze. You also get seeds without having to spend seeds.