I have found Coinbase to be pretty convenient for buying and selling bitcoins.
For how to keep them secure once you have them, I recommend simply doing some research. Coinbase provides a basic wallet and if I remember correctly, they cold store most of their coins offline.
Blockchain.info is another good option for a basic wallet. It's simple with a good amount of features. Your wallet is also decrypted in-browser and not on their server so they don't store compromising information.
There are also concepts such as a paper wallet [1] and a brainwallet [2] for extreme security. A brainwallet is basically what it sounds like - a way of storing your wallet in your head.
"Sorry, the maximum number of purchases on Coinbase has been reached for today. Please try again in 24 hours. We'll continue raising this limit over time."
These sort of security and reliability concerns are certain to hold bitcoin back. Granted they also represent the opportunity, as surely they will be resolved.
Well there's always MtGox, which I don't think has any limits, but you have to give them an absurd amount of identity verification to be able to place orders, which scared me away.
There are multiple services, Coinbase is just one of them. This particular instance of you not being able to place an order through them is not indicative of a wider tendency to not be able to place orders.
Coinbase has a 24-hour rolling limit for how many coins can be sold through them. This is reliably the case every day.
Also, not being able to buy coins doesn't seem to be a "security" issue.
For how to keep them secure once you have them, I recommend simply doing some research. Coinbase provides a basic wallet and if I remember correctly, they cold store most of their coins offline.
Blockchain.info is another good option for a basic wallet. It's simple with a good amount of features. Your wallet is also decrypted in-browser and not on their server so they don't store compromising information.
There are also concepts such as a paper wallet [1] and a brainwallet [2] for extreme security. A brainwallet is basically what it sounds like - a way of storing your wallet in your head.
[1] https://en.bitcoin.it/wiki/Paper_wallet [2] https://en.bitcoin.it/wiki/Brainwallet