Im pretty newbie in cryptocurrency. I heard that we need to keep secure our Private key. Do we get a Private Key each time we open an account on an Exchange ie: Bittrex, Poloniex, Binance, Coinbase, etc?
You do not hold any private key in any exchanges. They hold the private key of the deposit address you use in each exchanges. That is why it is not advised to hold your crypto in any exchanges.
or do we get Private Key on each currency? I have account on Bittrex and Binance and Coinbase, and as I can remember they never show me Private key.
You only get public keys or address for each crypto where you can deposit your crypto.
I also need to know; Do we need to save the generated address each time we send or receive coins between exchange? the address keep changing each time we send.
Not necessary, just get the new one.
And where do I get a Paper Private key?
You can get the private of the address from the wallet where you generated the address. You can print it and store it safely where no one can steal it.