No, bittrex and any other exchanges are really bad to store any kind of coin. Such websites could be hacked or simply disappear with users'money, so don't trust them. Never.

You need to have full control of your coins. Think as cryptocoins as a "traditional" wallet for banknotes and coins. Do you give it to strangers? Of course not. You take care of your wallet by yourself. So you have to do with cryptocurrencies. No difference.

For altcoins it depends on which altcoin you wanna store. If they are ETH, ETC or Ethereum-based tokens you can store safely on myetherwallet. It is not a web wallet (although many still think so...) and you have full control of your coins. If you have other altcoins (for example Litecoin, Dash, etc.) you may use multicoin wallets like Coinomi (for mobile only...) . Anyway, before downloading a wallet check if it's safe to use, search for feedbacks and keep it safe and updated.