There are two ways to trade with other players. First is through the site's User Exchange, accessible from the Community dropdown at any time. You may also notice there is a Trade Hub, which is used to list items publicly that you wish to trade. In the Trade Hub, you can also view items other people are offering up for trade. The Trade Hub is only for listing, and not facilitating the trade itself. You will have to communicate with the other player and set up a User Exchange in order to trade.
Within the User Exchange, be sure to comment any details of the trade, and if you have a forum or even Discord DM message history, you can link it as the proof of the agreement. Otherwise, you can leave it blank, but understand that staff can't access your DM history if you need to provide any kind of proof later on. The User Exchange will walk you through the process of the trade, and you and your trade partner will both need to confirm the trade twice before it completes.
The other way to trade items is by gifting them directly from your inventory to the other player, by clicking the item, then the checkbox, select Transfer Item, choose their username from the player list and then click Transfer. The item will be instantly removed from your inventory, and will create a log in your history stating that you sent the item. This is not recommended for trading, only gifting.
You can also trade a character or geno from the User Exchange page. Secondarily, you can go to the character or geno page, where on the sidebar you will see a Transfer link. From that page, you can choose the recipient player, and you will need to give a reason in the relevant field. The recipient will have to accept, and then the character or geno will be transferred immediately. Make sure to double check the Design Terms of an existing import before trading, if it was not designed by you.