The most fundamental part of trading in Team Fortress 2 is the currency. In TF2, the currency is in 2 forms:
"Pure", which is entirely in Keys and Metal (more on that later), and "Impure", which is equating value and offering a certain value item for another (e.g. 100 key hat for a 100 key weapon).
Keys
Keys are by far the strongest currency in TF2. They have a real world value that varies slightly, but is always $2.49 from Valve, and around $1.89-$2.00 each when bought secondhand from sites like marketplace.tf. Also, not all keys are created equal. Crate keys are different, often worth more than one key each. When talking about keys, it is referring to a Mann Co. Supply Crate Key.
Metal
Metal is the equivilants to cents on the dollar. Refined metal is the largest, worth 3 Reclaimed metal, which is worth 3 Scrap, which is 2 weapons. One refined is worth 18 weapons. Typically, metal is measured in Refined, using the others as change. Each refined is typically worth anywhere from $0.02 to $0.04, and the ratio from keys to metal varies quite a lot. At the time of this website, a key is $1.93 and has the exchange rate of 74 ref.
Hats
There are a few constant hats, including Bill's Hat, Earbuds and more. They are hats that were limited time, but were common enough to have a price that stays consistent like the key. Nearly every other hat varies in value, especially ones with sought-after qualities. People also refer to all cosmetics as 'hats'. Cosmetics are probaby the most traded items, because unusual cosmetics or strange cosmetics are very popular.
Websites
There are many websites for trading in TF2. In order to calculate the cost of keys versus metal versus the dollar, one popular website is
calculator.tf. The downside of calculator.tf is that it is sometimes not fully accurate, and can give incorrect exchange rates. One of the most used trading websites is
backpack.tf. The major upside for this is that it is extremely popular (so you will get the market price), and that it is almost exclusively for trading. If you are looking to purchase something with real world money, then the best place to look at it
marketplace.tf. Marketplace.tf is almost entirely automated with official website bots, so you do not need to discuss prices or have sketchy trades. You can pay with card, paypal, and other options.
Websites to avoid
There are a variety of websites to avoid, for many reasons. Some of these include sketchy practices, rigged gambling, false advertising, or bad deals. Granted, some of them have upsides. One perfect example of this is
scrap.tf, because it has great automated trading for regular weapons that you get in drops. You can automatically trade weapon duplicates or pick and choose, and get the same metal as smelting them in-game. They don't even need to be the same class or anything. However, they purchase everything that is not a regular weapon for far less than they are worth, and sell them for far more than they are worth. Also, any gambling sites are scams. They give you borderline worthless crates and items while not giving anything worthwhile in return. One of these is tf2cases.com or unboxer.tf. They are complete
SCAMS, but pay people to tell fake testimonials or promise free crates. Remember: if it sounds too good to be true, than it absolutely is.