If I buy Bitcoins at Coinbase.com, should I leave them on Coinbase.com, or should I move them into a Bitcoin Wallet? Why? If I buy Bitcoins at Coinbase.com, should I leave them in my account at Coinbase.com, or should I move them into a Bitcoin Wallet? Why?
This depends solely on trust and ease of use.

Trust

Who do you trust more? Coinbase's security setup or your own? Some things to consider:


Coinbase makes profit from it's website and so wants people to use it. For this reason they benefit from taking security serious. They probably have a dedicated team thinking about how to improve the security of their system.
You own computer, let's say running Windows, is likely very much less secure. Any virus or malware can steal your coins. Also, without a backup, a disk failure will make you lose all your coins.
While Coinbase's security might be much stronger than yours, it is very more likely to be targeted by hacking attempts. They store a very large amount of coins, while you only store your own.


Ease of use

Running a Bitcoin wallet yourself can sometimes be some hassle, while e-wallets can offer you extended functionality and features.


You will need to download and sync the complete block chain, which takes around 9 GiB or storage. There are also clients that don't require you to do this, but still they have to sync all the time and use network bandwidth.
You will need to worry about backups. You can trust Coinbase for taking backups regularly. If you want your own wallet to be safe, however, you will have to backup them yourself.
Much e-wallets like Coinbase offer extra functionality, like links to information on blocks, transactions and addresses. Also many of them have a Mobile App that you can use to access your wallet with.