Yes, you need a wallet that supports the coins you purchase / trade.
Take a look at this lists, for which coins each wallet supports and what platform it is available for.
https://www.cryptocompare.com/wallets/#/overview
Never trust the exchanges to hold large volumes of coin for you, as they are not in your control.
Remember, always backup your wallet.
Heaps of tutorials and videos online.
Awesome, thank you very much!!! Its a minefield to work through. That is awesome. So I take it people have multiple wallets a lot
If you have many different coins, yes, then that might be the case.
There is the question of which devices you store the wallet on and whether you can sync those up. Not all wallets can sync across devices. There is where you have to research each one and whether it fulfils your needs.
Just take your time, read and take notes. It all starts to make sense eventually.
Oh, another tip, guard the keys to your wallet very carefully. Many people have either been tricked, or made silly mistakes, revealing their keys and some one has then stolen their coins.
Again, read up on this and inform yourself. In the crypto world, you are responsible for yourself.