There are a few different ways to buy bitcoin, but the simplest way is to use an online exchange. My personal favorite is Coinbase, which allows you to buy and sell bitcoin, ether (the currency of Ethereum), and Litecoin. You can also use Coinbase to store your coins in a digital wallet.
Another popular exchange is Gemini. Gemini is based in the United States and allows you to buy bitcoin with US dollars. Bitstamp is another option, but it's only available in Europe.
Once you have your bitcoin, you can use it to purchase goods and services online or you can hold onto it as an investment.