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When you find your first potential trade partner, click on the target city, and then click on the Diplomacy button.
I often find that having a nice letter pre-written and pre-formatted often gets a better response than something I wrote directly into the diplomacy window.
Adding a touch of roleplaying often gets players intrigued as well.
Make sure that your letter clearly states what you are offering and what you would like in return.
I always suggest that when initiating trade routes, request small trades first.
Between 300 to 500 units is a decent small trade.
This way, both sides feel as if they are unlikely to lose much should the trade go awry.

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Once you and your partner have agreed to trading terms, access your trading partner’s city by either finding it on the map or by clicking the name of the city in your diplomacy (letters) menu.
Then click on your trading partner’s city and pick the Transport Goods button.
This will take you to your trading ships menu, and you will be able to load your ship(s) with the appropriate goods.
Do this by either using the slide bar or by typing in the amount you wish to ship off to your new friend.
(I usually type in the amount.
The slide bar can be quite imprecise at times.)
Now you simply wait for your ship to reach your trade partner and for his/her ships to reach you. You can both track the status of ships coming into your docks by checking the Trading Docks. (Please note that incoming shipments will not show up until after the goods are done loading.) Once you both of you have received your shipments, your first trade is complete. You may now wish to attempt larger trades. Please trade judiciously and no more than you can afford to lose if you are trying to establish new trade partners.
Ikariam Brasil players should find the above works on their version of the game as well.