wallet security: I'm new to Bitcoin, how do I keep my bitcoins safe?

Use a desktop or mobile wallet (Electrum or Mycelium) to spend bitcoins (be sure to write down your recovery seed and store it as a paper wallet). To receive and store them, use offline paper wallets. As soon as a paper wallet exceeds a certain amount (50mXBC for example), save it and start a new one. If you are spending and your online wallet runs out, open a paper wallet to complete the transaction, the remaining coins will remain in the online wallet.
NOTE: with "online wallet" I mean wallets like Electrum and Mycelium, do not web wallets like Coinbase or the Blockchain.info wallet.
Someone posted instructions for paper wallets in bitcointalk:


Do your own research before risking your coins by following what someone told you.

Do not use web wallets.

EDIT: Brain wallets (memorize a passphrase that is encoded to produce keys and addresses) are very controversial in the bitcoin community. The correct way to use them is to choose the passphrase evenly and randomly of a large set (at least 2 ^ 128).
Security enhancements include:
Add salt (something unique but trivial to remember, such as your full name, phone number or email address) to the passphrase to prevent hackers from attacking everyone at once.
The use of WarpWallet key stretching is an algorithm that derives keys from a passphrase using a difficult algorithm that requires a lot of memory, this makes them a lot harder to hack.

To generate multiple wallets from a passphrase, you can attach an index (add a number to the passphrase).

The main advantage of brain wallets is that there is no physical wallet to steal or confiscate, and the plausible denial of having bitcoin.
The disadvantages include the difficulty of remembering the passphrase (if you forget it, the coins are gone), the possibility of theft (there are hackers who constantly monitor the addresses created from weak passphrases and steal coins immediately) and the fact that in the unfortunate case of your death, your family could not recover your coins.
If you decide to use them, they can be used instead of the paper wallets in the first paragraph.