Crypto wallets keep your crypto assets safe and fully under your control.
You will need to download a wallet that can hold your crypto - It may be a good idea to look into this before you buy a lesser known coin.
There are many wallets out there, some are for specific coins, others hold multiple cryptos.
Once downloaded your wallet you will need to sign in and recieve your private keys - These are the most important thing!! Never lose your private keys! Lose them and you lose your crypto permanently!!
Keeping your assets on an exchange is not deemed best practice as you don't have the private keys for your exchange account. Therefore, if the exchange closes/disappears, or is hacked, you risk losing everything you have on there. So, unless you are trading on the exchange, it is always advised to send your crypto to your wallet!