Don’t use exchanges to store your bitcoins for any length of time. Exchange storage is only as secure as the exchange’s security infrastructure, so although many people do use this option, the coins are still not within your control. Storing on an exchange should not be considered as anything other than a temporary option.
Instead, use a software wallet (such as the Bitcoin QT client) to store your bitcoins. A software wallet allows you to secure your bitcoins on your own computer. Encrypt the wallet and make backups to ensure your bitcoins are safe. This option requires you to carry out virus checks and have a good understanding of Internet security. Alternatively, you can try a popular online wallet such as the one offered at Blockchain.info while you become familiar with the workings of a wallet. This can simplify the process for you.
Another option is to use a paper wallet to send your coins to a bitcoin address that is not connected to any online exchange nor to software that is on your computer. This bitcoin can only be spent when you decide to manually redeem it through using your private key.