There are many ways to buy bitcoin online but the absolute easiest way to buy bitcoin is from Coinbase.
Coinbase allows you to buy bitcoin, ethereum and litecoin either directly from funds in your bank account, or from your credit/debit card.
They offer a simple sign-up process that offers 2-factor account authentication for your own security. While you will need to provide a photo-ID in order to purchase bitcoin, this is only for your protection.
Note that Coinbase does offer a small fee for purchasing and selling bitcoin.