Keeping currency on exchanges - It's best practice to transfer your currency to storage that YOU are in control of whenever you've purchased it. This proves well to encourage long-term holding too as keeping money on exchanges may encourage constant price checking and selling rather quickly also. A good example of why you should expect the worst even from the best exchanges is the MtGox incident (feel free to Google it) personally I'd trust my funds on an exchange such as Binance or Kucoin, but I keep certain funds on desktop wallets and certain on paper wallets and some on exchanges for swing trading.

Recommended wallets - For Ethereum, I'd use MyEtherWallet, simple to get started with and relatively secure. See Metamask for an even securer alternative. For Bitcoin, Litecoin and everything else, I can only recommend hardware wallets as there have been many bad incidents that have happened to users with wallets such as Jaxx etc. For cheap and extremely secure hardware wallets, see both the Ledger Nano and Trezor, a bit costly but how much do you value security? Is the honest question.