You can see the "Trade" button above, alongside with the notifications and your inbox. In the trade section you'll find the most recent threads of other people with items being offered/searched for.
You can also see the auctions below. A user can place an item for auction, specifying a starting price. Whoever bids the highest amount, gets the item at the end. Each bid is a minimum of +5 tg above the current bid.
People usually trade with tg (trade gold), you'll hardly ever see somebody trading for specific items. This is just because the tg is neutral and can be traded for every item. You can sell the things you don't need for tg and then buy things you want with that tg.
Messaging is done either here through the forum (inbox), through discord or directly in game. By clicking on the account of someone (for instance when you are checking his thread), you can see his TSW (in game account) and use a link to message him on the forum.
Tg collection is the easiest at the beginning of the season (2-3 weeks from the start), because items like arcane crystals and signets are needed by everyone. All prices fall down with the progression of the season. You can still get some tg by selling acs and sigs, as well as some good late game items.