Where can I store my altcoins? I was thinking of buying and trading some altcoins, but can I just store them in like for example bittrex? Or should I download the altcoins individual wallet?
It's safer to use individual wallets where you control the private keys, especially if you have a large amount invested. Exchanges are risky because there is a history of them getting hacked, stealing coins, mismanagement, and then going out of business (MtGox, Cryptsy,etc.).
Only use exchanges for trading and then move them off of the exchanges. Sometimes holding is better than trading anyway.