Can trading be done on coinbase.com?
Coinbase unlike Mt.Gox is not a bitcoin exchange, it is a bitcoin wallet with the option of buying or selling bitcoins using a US bank account. You can store bitcoins on the site but not USD. 

This is primarily done for legal reasons. When you allow people to trade currency with each other instead of just being a currency dealer you open yourself up to a whole world of regulatory problems. Namely people would have to be extensively verified and new anti money laundering systems would have to be put in place to comply with FinCEN.