the vast majority of pictures online tend to be pretty inaccurate, and the ones that are correct usually just show your standard brown tunic.

What sort of pictures do you mean? They may actually be more accurate than you assume, because from "your standard brown tunic" it sounds as though you may be dealing with certain misconceptions about medieval dress. It's common for films that are setting themselves up to be "gritty" or "realistic" about the Middle Ages to dress everyone in shades of mud to show that the production understands how poor people were and how expensive dyes were. Highly-detailed and realistic illustrations from Books of Hours (religious texts for lay readers, which sometimes included sections for calendars with full-page artwork showing what noble and peasant individuals would be doing during each month) do show laboring peasants in greys and browns, but also in blues, pinks, greens, yellows, and other colors. Madder (for reds, oranges, pinks, and browns), dyer's weld (for yellow), and woad (for blue) grew natively in England, although for dyeing on a commercial scale woad and madder were typically imported from the continent.

So I'm not saying, "sure, just take all pictures you see at face value!" But you may (may) be coming across images of clothing made by artisans who really know what they're doing - the SCA gets a bad rap, but some SCAdians do incredible jobs replicating period fabric and clothing, e.g. - and I'd hate to think they were being dismissed automatically because they don't conform to the pop cultural ideal of the medieval peasant outfit.

But what if I'm a peasant in 14th century England during the winter? What materials would I be wearing? Fur, leather, etc? A hat? A hood?

So, some of the best sources for this kind of thing are books of hours that hew to some degree of realism. One such is very early 15th century Les Très Riches Heures du Duc de Berry, illuminated by the Limbourg Brothers, which has beautiful pages for each month, most of which show the activities of the peasantry as they work the land, even in the winter (apart from January). November has pig-tending in the woods, December has hunting, January a great feast, February chopping wood, and March tilling the dry fields.

For your 14th century English peasant, the name of the game is wool. Wool was a major industry in England and the Low Countries, and while today we think of it as an inherently heavy, dense fabric, it could be made in a huge variety of weights - so it was worn through the year. It could still be made as a heavy, dense fabric, though, and in fact it was often fulled (slightly felted) to the extent that the raw edges didn't have to be protected to keep from unraveling the way that we treat hems and seams now. A peasant would wear wool of the lowest grades, made of the fleece from lower parts of the sheep; in the cheapest cases, the wool would be an unbleached/undyed grey like that of the winter-est peasants in the Très Riches Heures (another reason why mud-brown tunics are not authentic). This means a wool tunic/cote, wool supertunic/surcoat, and wool stockings. These stockings would be cut out of fabric, rather than knitted - although knitting was known at the time, stockings were typically cut out of cloth on the bias (diagonally across the weave) in order to have some stretch and sewn to fit close to the leg. A cloak or just a blanket wrapped around the body, made of a wool too dense to wear, could also be used for more protection.

A hood would definitely have been worn, as hoods were a very common accessory in the 14th century. There's a great image of the pattern shape of an ordinary hood in Textiles and Clothing, 1150-1450: straight front, loose-ish neck and cowl (which you'd certainly want to protect you against the winter winds), and long liripipe (the thin tube that hangs off the back of a stereotypical hood, as seen on the woman in blue here).

Fur was unlikely. Since it doesn't survive the way wool, linen, and silk do, we have much less information about it than we do textiles, and we're left with textual sources. These mainly discuss the expensive furs that would be used to line outer garments for the wealthy, like squirrel, sable, and ermine. Rabbit may have been the cheapest source of fur, and for use as a lining you'd have to a) get a lot of rabbits and b) put a lot of effort into sewing their skins together, which is part of why fur in general was expensive. I'm not going to rule it out because, like I said, I don't know as much about fur, but it seems unlikely. Lining boots and gloves, though, where a single skin would be enough for each, is more plausible.

The first thing to note is that there wasn’t one homogenous peasant class. There were “rich” peasants who had managed to acquire a lot of land, and thus extra income and purchasing power, and there were dirt poor peasants without enough land even to support themselves; these latter hired themselves as servants to other peasants or were reduced to begging. This means that more well-to-do peasants could wear better quality clothing that poor ones. There would be fewer holes and less patching in their clothes and some articles may even be made from material bought at market rather than homespun. Some of these may even have some color instead of being the drab color of undyed wool, called hodden gray especially in Scotland, which you mention as typical of peasant dress. Most peasant dress would be woolen and woven at home on looms. The iconography of a medieval woman was a figure holding the distaff and spindle used to spin thread.

The second thing to note is that there is very little winter dress per se. It was more a matter of putting on more layers of everyday dress than breaking out the winter gear. One exception to this is woolen mittens, which were characteristic in depictions of 14th-C English peasants in winter settings. These were stitched with 3 compartments, one for the thumb and two for the fingers, and appear a few times in the famous Luttrell Psalter from the mid-1300s, as in this example (though it would be very unusual for them to be particolored as in the illumination).

As a peasant, you better not be wearing fur. “Sumptuary laws” began to appear in the late 13th C legislating who could wear what; these were meant to make the rigid medieval social order visible. One law from 1337 limited cloth made on the Continent to the royal family and ordered that fur could only be worn by “the King, Queen, and their children, the Prelates, earl, barons, knights and ladies and people of the Holy Church” (Baldwin, 31). (The 14th-C demand for fur is staggering: In 1344-45 Edward III’s household apparently needed over 79,000 miniver skins.) A sumptuary law of 1367 ordered, among other things, that

Carters, ploughmen, drivers of the plough, oxherds, cowherds, shepherds, swineherds, deys, and all other keepers of beasts, threshers of corn, and all manner of people of the estate of a groom, attending to husbandry, and all other people that have not forty shillings of goods nor of chattels, shall not take nor wear no manner of cloth but blanket and russet wool of twelve pence, and shall wear their girdles of linen according to their estate . . .

Headgear could be various. A peaked, pointed hat, sometimes shown worn backwards—like a medieval baseball cap!—was a typical hat shown in manuscript illuminations, but so was the classic “cucullus” or hood, derived ultimately from the monastic cowl. If you imagine a modern hoodie and cut it off at the shoulders that’s basically the medieval hood. The peasant wearing mittens in the picture above also wears a hood. (The late Michael Camille warns us in Mirror in Parchment: The Luttrell Psalter and the Making of Medieval England to be cautious at taking the illustrations in medieval manuscripts as invariably accurate depictions of actual clothing, especially peasant dress, which is always idealized.)

Leather would be used for belts, probably for work gloves, and definitely for shoes.

Sources: Herbert Norris, Medieval Costume and Fashion (1927) is very useful for basic descriptions and nomenclature. Frances E. Baldwin, Sumptuary Legislation and Personal Regulation in England (1926); Margaret Scott, Fashion in the Middle Ages (2011).