As soon as you have a Vault in your village you can start trading resources.
To do so deposit the resource you want to trade into the Vault, go to Trade and enter your order.
Once the trade is completed the resources will go into your Vault (if your Vault does not have enough capacity to hold all traded resources those will overflow into your City Center but keep in mind that if your City Center is topped out by the overflow your resource generation ratio will drop to zero).
The new system allows you to trade at any time regardless of open orders but follows a supply and demand schema meaning that the ratios vary dynamically during the tournament.
The ratio is calculated the following way:
Initial pool of resource desired / Initial pool of resource offered + Amount of resource offered
The initial pool of resources is the following: