Trading is basically entirely 3rd party(well it's run by GGG but you have to get out of the game). You use
Trade - Path of Exile search for what you want with the many many filter options, then it'll display all the items that fit and their prices. You click the whisper button and it'll copy a premade message in your windows clipboard, then you switch back to the game, open a chat window and ctrl v paste the message and it'll send the person an automated messages like Hi, I'd like to buy your x for y currency. Generally the only interaction past that is either silence(guy is afk, is a bot, doesn't want to sell, whatever) or the guy will send you a party invite. Accept the invite, right click their portrait and select "Go to Hideout" to go to their hideout. Open a trade with them or wait for them to open trade(I personally wait since most people will be looking at their shit to find the item), trade the currency you said you'd pay(prepare this shit beforehand if possible, also you can't split currency once the trade window is up), make sure to double check what you're buying is what's in the window and done, leave the party and go back to whatever.
It's a pretty annoying process but if you don't trade too often, it's not so bad to do and once you're used to it and the seller/buyer isn't a retard, it takes like 20-30secs total(with a SSD).
To sell, the easy way is to buy a premium stash tab, which can be set as a sale tab, where you select the items and fix a price, and they'll be picked up by the trading API the websites use. However if you don't want to spend money, then you need to install a 3rd party program like Acquisition, which will list your items in normal stash tabs on the trading forum, which is also picked up by the trading API(but slower iirc). You could technically do the forum posts manually if you don't trust the 3rd party programs, but it's even more a pain in the dick to do.