To own cryptocurrency you must have a wallet.
While it may be easier to leave coins with an exchange or other administrator, it's typically not recommended. Many exchanges and online wallets have suffered from security breaches in the past, using such a service is placing trust in them. If you choose to trust an administrator it would make sense you ensure a return on the balance.
A good wallet is private, open source, and well tested
A private wallet is one where only you have the private keys. This is most similar to holding cash in your hand. It does not mean it’s invulnerable to theft, it should be treated much like cash. Seeing your private keys is sometimes not an option from within the wallet software, so how can you know you have the keys? Most private wallets will prompt you write down a phrase of 12 or more words as a backup, this is known as a mnemonic “seed” and is capable of restoring your wallet even if your device is lost or destroyed. Since it is a backup of the private keys it will even restore crypto that is added to your wallet after making the backup. The seed phrase is your wallet, and it isn’t normally protected by a password and is not protected by a lock screen, it should be stored somewhere very safe and never stored on a phone or computer without expert precautions – do not screenshot or print seed words, write them by hand or use a metal storage device. Jameson Lopp periodically publishes stress tests on these devices which should be designed to survive a house fire.
There are many different kinds of wallets, including free software and hardware/offline wallets. It is extremely important to verify the wallet you intend to use. For new users this is likely only using a wallet found in the Google Play Store or Apple Store, and verifying positive reviews and a long track record of at least 6 months, beware of fake reviews and new wallets.
I highly recommend the following wallets:
This is a widely trusted wallet and the oldest lite wallet (meaning you don't have to run a full node). It's available for desktop and android. I use this wallet on desktop and paired with cold storage devices or multisignature. It has a lot of advanced features and compete control of your coins, but also can be used in very basic setups where it's quite simple.
This wallet has great features similar to Elecrum but is highly functional on a mobile device. Available for both Android and iOS this wallet is recommended by many seasoned bitcoiners. It can also be used in conjunction with Elecrum to create multisignature wallets.
A bitcoin only hardware wallet with advanced features, including air gapped setup and use.
A fully open source hardware wallet supporting Bitcoin and altcoins.