First of all you will need a Wallet - no worrys - you dont need to buy -
there are few options how to store your private keys - the coins are all time stored in the BlockChain - the Private Key is the Key which you need to Save and hide. You will need this Key to send Coins out of your Wallet.
Your Wallet :
once you have created your Wallet you are Ready to Buy some Coins.
you will get a Private Key which looks like "e15da7a1236855c6317e5ef510ee135ab4e71812a29e58c54508970bdcb6b842"
and you will get a Public Key which looks like "0xc257b170b6D3A0C248cf7d6508fe6624B1f63C7b"
the Public Key is your Bank Account - the Private Key your Pin - the Only Diffrence to this is that the Private key includes the Public Key
if anyone knows your Private Key they can Steal your Money !!!
and there is no way back !!!
the eXchange changes your Fiat ( Current money you have on your Bank Account ) to a CryptoCurrency
i use therefore Kraken.com you will need to create a Account and fufill the KYC procedure ( KYC - Know Your Customer ) - once done you are Ready to go on.
Send Money from your Fiat Bank Account to your Kraken Account. Do not trust any Agency which wants to Buy Coins for you !!! if your Money is on the Kraken Account you can directly start Buying.
once you have buyed Ethereum and you want to keep them longer send them to your Wallet
when ever you will send Ethereum from your wallet back or somewhere else you will need your private key.
may it comes the meaning that this would be not save enoug. but shortly explained.
how to generate a Private Key: take a Coin and mark one side with a 1 and the other with a 0 than flip him 256 times and note all time the 1 or the 0
thats finaly your private key in Binary form. this one will be calculated into a HexaDecimal String- to solve this you will need to try it 1081 times.
even the fastest computers this days would need a few billion years to solve this