To trade in MTGO, you could go to the Classifieds section to see all offers. Or you could find some good bots (basically automated stores) to trade with. Another option is to type "/join auction" in a chat window.
You can initiate a trade with someone by (right)clicking on the name and select Trade. Make sure the cards you wish to trade are actually made tradable in your Collection.
If your trade, you can see their tradable Collection and select the agreed cards (or with a bot, the desired cards) and click on the Confirm button above the trade-binder. You will then see a summary of the trade, to check if it's okay. Confirm again to get the cards.