Well simply put trading is the act of in game swapping of items.
In order to trade then
Load Team Fortress 2
Go to the "Items" tab (whether you're on a server or not)
Click "Trading"
You can
1) Trade with anyone on the sever you are on
2) Trade with friends whether you're on the same server or not
3) Trade by copy pasting a person's Steam URL.
2 Weapons of any specific class can be crafted into 1 scrap. The items must be of the same class, so both Medic weapons. It doesn't matter what weapons, as long as they go to the same class. If it's a Medic and a Scout weapon for example, the game won't let you scrap them out. If you run into a player saying "Scrapbanking", that means that you can trade them two items of two different classes (such as 1 Scout Weapon and 1 Medic Weapon) and they will take the items for you and give you a scrap.
3 Scrap can be crafted into 1 reclaimedou use
3 Reclaimed can be crafted into 1 refined.
If need be, the process works backwards. You can smelt 1 ref into 3 rec, and 1 rec into 3 scrap. You cannot get back any weapons used to make the scrap however. Also, if you use an item that says in the item "Achievement Item: Not Tradeable", this means that if you use it in the scrap, that scrap cannot be traded. An item that is non craftable will be dark and unusable in crafting.
Pricing works as thus
0.11 = 1 Scrap
0.22 = 2 Scrap
0.33 = 1 Rec
0.44 = 1 Scrap and 1 Rec
0.55 = 2 Scrap and 1 Rec
0.66 = 2 Rec
0.77 = 1 Scrap and 2 Rec
0.88 = 2 Scrap and 2 Rec
1.00 = 1 Ref
Keys are the second type of currency, their prices fluctuate but a lot of people only trade items for keys. They have run anywhere from 2-2.66 ref since I've started playing.
Buds and Bills are the third type of currency.
Buds stand for Earbuds, which are worth large amounts of Keys. Bills known as Bill's Hat are also worth a lot of keys. Both Buds and Bills are items that can be equipped to your character, but a lot of people only trade these.
Also prices are very important.
Any standard item (yellow text) is worth 1 scrap at max, unless it's special like the frying pan. All standard items can, for the most part, be obtained while playing, as the game will randomly toss you items just for playing the game. These drops happen at random, and stop when you have accumulated 11 hours of play. This 11 hour timer resets every single wednesday, allowing you to collect even more items.
Any vintage item or genuine item (blue text and green text respectively). Vintages are extremely old items, and genuines are special pre order items. These items are rarer and require more metal, but are the same as the standard weapons in performance.
Any strange items are expensive, and can track the number of kills you've acquired on that weapon. The kills are cosmetic, as the weapon performs just like the standard. Stranges can only be obtained through crates (which require keys to open).
Unusuals are extremely expensive items that are animated and have special appearances. They're extremely interesting, but usually cost several buds.
People tend to use three sites for prices
http://tf2spreadsheet.blogspot.com/ which contains a comprehensive search-able spreadsheet to read.
http://www.tf2mm.com/ which is a place where only metal exchanges happen. You can post what you're selling for what price, or what you're willing to buy for a certain price.
http://www.tf2outpost.com/ where people can post what they're selling and what they're buying for certain prices.
Anyways, hope that helps. Enjoy.
As for server requests, just click some games in the "Start Playing" button, or throw caution to the wind and try random servers. Honestly, my personal favorites are Darkly Trade Servers or Control Point Maps. Even on a trade server, you can run around and kill people for fun too.