Hover over "Recent Activity", and click "Recent Trades" or "Recent Challenges" (These can also be accessed through the Pokémon Center).
In the trading page, there are six boxes. The first is where you choose the offer Pokémon - Click one of the numbered buttons to show some of your Pokemon (Ordered by Pokédex No., changeable using the "Sort" box at the top of the box), and click the radio button underneath the one you want to trade. In the second box, scroll down the list to find the Pokémon you want in return (Ordered alphabetically). The level box is where you choose what the minimum level of the return Pokémon. Then click "Submit Trade Offer" and wait for the trade to go through!
Right now I can't make a trade... It says "That Pokémon is not yours!". Odd.
On the challenges page, there are two tables. The first shows your challenges, challenges from others, and previous challenges. Pretty self-explanatory. The second table is where you challenge a friend. In the first box, there's a space to write the name of the person you want to challenge (Or you can click Random Challenger, but I don't recommend it due to inactive members). Then you choose the amount of money you want to bet, then you choose which Pokemon you want to use (Same format as trading). Then click "Make Challenge", and wait. Battles are done automatically, you and the challenger both pick a Pokémon, then a battle's generated depending on type and level. You can have a look at a battle once it's finished by looking in the "Historic Challenges" box.
