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.
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.
Coinbase doesn't actually run an online wallet. They store your coin in their wallet, and they hold the keys to your money. You are trusting them to not run a fractional exchange, since they don't publish BTC assets & liability trees. Personally, I don't think that's secure at all.
You should store the bulk your coin in your own, personal wallet with a big nice long password, or even better, with a generated password (passhash.com for example). Electrum, CarbonWallet, Armory all make GREAT secure "deterministic" wallets.
Make a "seed backup" of the wallet, and keep it in a safe place. You can print a a backup without a password if you're worried about losing the password, and lock it up.
Keep a second "spending" wallet around with less money in it than your main one. Maybe a simple password to type out at a cafe, etc. That wallet should also be backed up and imported into your cell phone (Mycelium) wallet. Then you can easily move funds as needed from your "big" wallet that's more heavily encrypted to your cell phone one.
There is no reason to use Coinbase as a wallet just because you use Coinbase as an exchange.
Coinbase does provide an online wallet and if you want to keep things simple, feel free to use it. But your question really boils down to which wallet should you use, since there is no significant advantage to using a wallet just because it's provided by your exchange other than reducing the number of passwords you have.
Steven Roose's answer goes into the details of whether you should choose an online wallet (similar to Coinbase) or run a wallet on your computer. If you decide to use an online wallet, Coinbase is just one of several choices. Coinbase's wallet is very simplistic, but if you are new to Bitcoin you may not care. Coinbase's wallet does have the advantage of allowing you to make instant purchases at places that use Coinbase's payment system.