What do you think about keeping the coins in Bittrex? Is it safe or should I go download wallets on my pc?
Nothing other then your own wallet is safe, remember that. You should
always download your own wallet, there is no reason not to. It completely misses the whole point of Bitcoin if you don't have a local wallet, you gain to benefit from Bitcoin's security nor privacy.
If you really need to trade on these centralized exchanges, then fine, do the trade and get your funds out. These exchanges were hacked multiple times and they will get hacked again, it is just a matter of time.