The following is true for fabrics from the late Bronze Age until the Industrial Revolution.
When choosing a fabric, four parameters have to be considered: Fibre and weave, colour and pattern.
First you need to decide whether it's only supposed to look authentic from afar, from a few steps away, or be as authentic as possible. In the first two cases, almost all fibres are fine, even if they're synthetic, if the other three parameters are authentic and if the fabric doesn't look artificial. Most people find it easier to judge the weave while ascertaining that a fabric is made of silk rather than, say, rayon often requires a burn test. But if you're serious about authenticity, even the fibre must be one out of four...
Not all synthetic (=man-made) fibres are actually plastic: Rayon and acetate are made from cellulose, i.e. from plant fibres, but they are nonetheless artificial in that they require a chemical process that was only discovered during the Industrial Revolution. The oldest technique for making artificial silk dates from 1885.
Linen and Wool are the oldest fibres in Western and Central Europe. The fibres of nettles were probably also used early in history, but what is nowadays sold as nettle-cloth is usually made from cotton. Silk was processed from the early Middle Ages on, but remained astronomically expensive for centuries. Around the same time, Cotton was cultivated in the Near/Middle East and imported to Europe as early as the high Middle Ages, but it remained a luxury good until the 17th century. It was woven on a linen warp in order to make it cheaper*. In countries that had strong ties with cotton-growing regions (Egypt, India, southern North America), cotton became more affordable relatively early (17th century), i.e. somewhere between wool and silk but still more expensive than linen. (I am referring to Europe here - I am told that in the 18th century, cotton was actually cheaper than linen in North America.) During the first half of the 18th century, cotton was forbidden in England, France and Prussia. When the prohibition was lifted, it became one of the most popular and affordable fabrics next to linen.
In the course of the 19th century, cotton gradually became more pupular than linen. While linen was still the first choice for underclothing, table and bedclothes until the early 20th century, it was completely superseded for upper garments: Cotton is easier to dye, doesn't crease as much, could be made into shiny threads by mercerising (since 1850) and is warm and dry in winter. Linen, on the other hand, is clammy in cool weather because it readily absorbs moisture, which is why it's wonderfully cool in summer and was popular for hand and dish towels up until the 1930s. The Industrial Revolution along with slave labour on cotton plantations in the South of North America made cotton cheaper than linen so that cotton gradually superseded linen even in those last three areas.
Silk was known from at least the Iron Age onwards. I don't know what type of silk. Princess Arnegunde (+ about 565) was buried with silk clothing.
Dupioni silk is one of the cheapest and most readily available kinds of silk today, which is why many people ask since when it was used. Well, I don't know. It isn't mentioned in any of my books and fashion journals right up into the 1960s and I know of no historical garment that is made of it. One source I've found says it was introduced in the 1940s, but I'll take that with a pebble of salt until it's corroborated further. One reader (thanks, Joan!) has, from personal experience, testified to its existence in the 1950s. By definition, Dupioni is made of a thread that owes its irregularity to the fact that two worms have spun one cocoon together, which is a rare occurrence. If you look at the supply in shops, about 80% of silkworms seem to have made that mistake. So obviously, Dupioni must nowadays be made "artificially". I suspect it's made out of the low-quality parts of the cocoon and was cleverly marketed in the course of the 1980s "back to nature" rustical trend. It's irregularity went well with that and contrasted nicely with the too-perfect-looking synthetic silk imitations of the 70s. Until then, any silk fabric as irregular as that would have been regarded as low quality, not even fit for lining.
Which is exactly what you should regard Dupionu silk as.
There are three main weaves: Linen (aka tabby), twill and atlas. Linen and twill were definetely known in the Bronze and Iron ages. Atlas probably wasn't used until the high middle ages. Twill weave can easily be made into a herrringbone-like pattern or into diamond twill. Both those patterns seem to have been popular during the early middle ages.
I can't give you any hints on which shops to check out: That varies from country to country, after all. However, there is one chain that can be found in many countries nowadays which sometimes carries suitable fabrics: IKEA. I guess that even the product names are the same in all countries. Bomull, for instance, is my favourite mock-up fabric because it's so cheap, and if the mock-up actually works, it can be re-used as lining. Some years ago, they had a fabric in tones of blue that re-created an 18th century Swedish fabric. Then there's Alina, the cheapeast linen all around which is suitable for lower/middle class upper garments, aprons and such. Right now, they carry three fabrics with patterns suitable for the 18th century and two or three suitable for the 19th, not to mention the solid-colour and striped cotton fabrics.
Be cautious around upholstery/window drape fabrics: They often sport suitable-looking patterns, but most of them are synthetic and depending on the era you want to portray, they may be too heavy. Up until 1800, fabrics usually were heavier and stiffer than what we know now, but not quite as heavy as upholstery fabric. Moreover, the patterns may be in the style of the period, but represent the period idea of upholstery or wall coverings. A good example is Toile de Jouy, a certain kind of cotton print that was produced in Jouy-en-Josas near Paris from the 1760s on, usually depicting pastoral scenes in one colour (red, blue, green or black) on an écru ground. Fabrics in that style can still be found among deco fabrics nowadays; they look period and actually are period - just not for clothing. They have been meant and used as home deco fabrics from the very beginnig and have never been used outside that context.
If you make historical costumes often or plan to do so, it is a good idea to cruise the fabric shops near you on a regular basis and buy whenever a suitable fabric is offered at a bargain price even if you have no immediate use for it. This requires that you have at least a vague idea of what you plan to sew during the next few years (yes, years: Fabric doesn't go bad) and how much fabric you need for each garment. For instance, I know that I need about 10-12 m of fabric for a complete 17th/18th century robe, so if I come across a suitable fabric at a good price, I buy 10-12 m of it. If my estimate has been too generous, I can always make jackets, corsets, aprons or patchwork out of the rest.
Whenever you travel, check out the places where fabric is sold. This should be especially interesting in countries where silk is produced, e.g. India, Southeast Asia, China. Even in the rich enclaves of Hong Kong (Western Market) and Singapore (Arab Street, Chinatown), silk is usually cheaper than anywhere in the "West". Be careful, though: It is often a good idea to do a burn test. An acquaintance brought back samples of alleged Thai silk from Thailand that all proved to be plastic when burned.
Usually you are told to either prewash or steam all fabrics because they may shrink - and woe if that happens after sewing! In most cases, this is good advice, but it depends on what fabric it is and what you intend to do with it. The modern books which recommend this have a modern perspective, after all. In fact, most are even wrong in some respects.
There are two things to consider: 1.) if and how much the fabric shrinks (quantity) and 2.) whether washing changes important properties of the fabric (quality).
If the fabric shrinks in the wash, that's not much of a deal because the loss is usually only about 10% and I expect that you'll have bought a little more than necessary anyway. What's more interesting is the question whether important properties of the fabric are changed and how. Since not only the fibre, but also the weave and dyeing process play a role, the best method is simply to try and see. In the case of many silk fabrics, for instance, the glue which gives the silk its stiffness and lustre is is lost. Stiff, rustly silk fabrics such as taffeta and dupioni should never be washed: They'd be limp afterwards. Taffeta tends to crinkle so much that even the hottest, steamiest iron is powerless. Since the Indian fabric colours apparently aren't all that fast, the fabric may lose a lot of its coulour. A dark blue saree fabric I own emerged from the tub royal blue. That's not altogether a bad thing since the colour may otherwise be dissolved by sweat and soak into whatever fabric layer is next to it.
In the case of wool it depends on whether the surface is meant to be smooth or felted. If you wash a smooth wool fabric, its surface may or may not felt; if you wash a felted fabric, it may felt a bit more - or not. I own a fichu made of very fine, smooth wool that has been in the normal (40°C) wash a couple of times but still hasn't felted. The knit stockings have been in the normal wash, too, but I haven't noticed any shrinking. I was told that quick temperature changes promote shrinking and felting, but I haven't tested that yet. Felting as such isn't a bad thing unless you want a smooth surface - quite contrary: Felting makes a wool fabric more impervious to cold, heat and rain. So while modern books tell you to treat wool fabric with great care, I say: Toss it in with your everyday stuff.
Linen and cotton tend to shrink (cotton by up to 10%, linen by less) but don't change their basic properties. The main problem is getting the creases out afterwards. Hang them up to dry when still relatively wet and iron them while still moist. Linen tends to be a bit stiff when dried out, but becomes softer when ironed.
To sum it up: Felted wool, cotton and linen should be washed at 40°C. In case the cotton/linen is meant for underwear, or if you don't know yet, go for 60°C. For smooth wool and silk, it's best to toss a piece in with the normal 40° wash (neaten the edges befor you do that!) and see whether and how it changes. Taffeta, Dupioni and Satin shouldn't be washed at all.
Modern books construe wool and silk to be very delicate fabrics that need special treatment and should only be ironed at a low temperature. IMHO that's bloody rubbish: Only plastic can't withstand high temperature. My own irons have always been set to the highest level and never manged to harm wool, cotton, linen or silk. Where ironing is concerned, I go by the Bavarian adage that "a good one endures it and a bad one perishes".
*) According to some sources, the motive was not (or not only) cost but the fact that cotton could not be spun into strong enough threads to endure the strain put onto the warp. That's quite plausible if you consider that sewing thread could not be made out of cotton until around 1800 for the very same reason. Tensile strength is not one of the virtues of cotton.