You can store your coins either online with the disadvantage they may be stolen if the respective web server is hacked or you can store them directly at your computer/cell phone etc. Third alternative would be to print the coins/keys ("cold wallet").

I´d recommend to use a wallet directly on your computer, e.g. bitcoinqt. However bitcoinqt downloads the whole blockchain (a list of all transactions ever made). Thats about 10GB. Therefore, I use electrum, which is also a wallet that stores your coins on your computer but does not download the blockchain.

http://electrum.org/