Read about technologies on bitcointalk.org, reddit, or elsewhere before trying it.
I’ve heard good things about hardware wallets, and would feel comfortable using one of the popular ones, like Trezor.
I have a lot of confidence in the Electrum wallet, which can live on your computer, and in the Blockchain.info wallet, which lives online, and you connect to it through a web browser.
If you use an online wallet enable two-factor authentication, so that your coins aren’t vulnerable to a stolen password.
For small amounts, it’s okay to leave coins on the exchange where you bought them. But know that exchanges are notorious for getting hacked or shut down — though the bigger ones (Kraken, Bitfinex, BitStamp, CEX) seem to have their act together.
For medium sized amounts, consider getting a dedicated computer. Install a clean operating system on that device, and only use it for your biggest investments. Do NOT experiment with any obscure crypto currencies on that device. And do not use for anything except your significant crytpo finances.
For large amounts, use an off-line paper wallet. Practice creating and recovering one first. Electrum allows you to do this for Bitcoin. It gives you a 12-word seed that can generate your private keys. So you install Electrum, record the 12 words, save one or more addresses, erase the wallet, then send anything you want to the addresses, and you can recover it at a future date using the 12 word seed.