For bitcoin, since I am from the Philippines I use Coins.ph mostly. But when storing BTC I use Electrum, they said that mycelium is a lot more simplier though I have not really tried one. BTW, both of them has there own private keys so I think whichever you use your bitcoin is safe.
In ethereum I use MyEtherWallet and avoid the application in playstore. Be sure that when choosing an ethereum wallet you own the private key. I was so dumb to use an app wallet in playstore without a private key. I have used it receiving a token and I cannot retrieve it now. MEW it is.
Is bittrex safe with all securities?
At the present time now, I would say yes it is safe now. But I just don't know how long it will be safe, like what I said it safer than the other exchange platform. So, you can keep your coins in bittrex for the mean time.
I think it will be much better if you hold it in a storage that you own the private key, apparently keeping it in exchange site is detrimental for your coin. Exchange sites are very much susceptible in hacking and things as such.