The first step is to set up a wallet to store your bitcoin - you will need one, whatever your preferred method of purchase. This could be an online wallet (either part of an exchange platform, or via an independent provider), a desktop wallet, a mobile wallet or an offline one (such as a hardware device or a paper wallet).
Even within these categories of wallets there is a wide variety of services to choose from, so do some research before deciding on which version best suits your needs.
You can find more information on some of the wallets out there, as well as tips on how to use them, here and here.
The most important part of any wallet is keeping your keys (a string of characters) and/or passwords safe. If you lose them, you lose access to the bitcoin stored there.
2 - open an account at an exchange
Cryptocurrency exchanges will buy and sell bitcoin on your behalf. There are hundreds currently operating, with varying degrees of liquidity and security, and new ones continue to emerge while others end up closing down. As with wallets, it is advisable to do some research before choosing - you may be lucky enough to have several reputable exchanges to choose from, or your access may be limited to one or two, depending on your geographical area.
The largest bitcoin exchange in the world at the moment in terms of US$ volume is Bitfinex, although it is mainly aimed at spot traders. Other high-volume exchanges are Coinbase, Bitstamp and Poloniex, but for small amounts, most reputable exchanges should work well. (Note: at time of writing, the surge of interest in bitcoin trading is placing strain on most retail buy and sell operations, so a degree of patience and caution is recommended.)
With the clampdown on know-your-client (KYC) and anti-money-laundering (AML) regulation, many exchanges now require verified identification for account setup. This will usually include a photo of your official ID, and sometimes also a proof of address.
Most exchanges accept payment via bank transfer or credit card, and some are willing to work with Paypal transfers. And most exchanges charge fees (which generally include the fees for using the bitcoin network).
Each exchange has a different procedure for both setup and transaction, and should give you sufficient detail to be able to execute the purchase. If not, consider changing the service provider.
Once the exchange has received payment, it will purchase the corresponding amount of bitcoin on your behalf, and deposit them in an automatically generated wallet on the exchange. This can take minutes, or sometimes hours due to network bottlenecks. If you wish (recommended), you can then move the funds to your off-exchange wallet.