Yes, of course. You can buy bitcoins from either exchanges, or directly from other people via marketplaces.
You can pay for them in a variety of ways, ranging from hard cash to credit and debit cards to wire transfers, or even with other cryptocurrencies, depending on who you are buying them from and where you live. Next, you will need a place to store your new bitcoins. In the bitcoin world, they're called a 'wallet' but it might be best to think of them as a kind of bank account.
Depending on the security levels you want, different wallets will provide different levels of security. Some act like everyday spending accounts and are comparable to a traditional leather wallet, while others tout military-grade protections.