Once you buy some coins, you are going to want to keep them safe. If you plan to hold them for a long time then your best option is a paper wallet. A paper wallet is essentially a print out of a private key and your bitcoin address. As seen in this paper wallet sample below, in addition to text, QR code versions of the bitcoin address and the private key are included so you can scan them with a cell phone or webcam easily.
Your bitcoin address is used to look up how many coins are in the wallet and to send coins to it. If you type a bitcoin address into a block explorer website like this one or this one you can see its current balance and any transactions that have ever been done with it. This information is public information that is kept in the public ledger called the blockchain.
Your private key is the passcode to your funds. If someone gets access to your private key, they can steal your coins. If you lose the private key then you lose all access to your money. It will be lost forever.
The most popular program used to generate paper wallets is BitAddress.org. If you didn't care about security at all, creating a paper wallet is as easy as going to that website, moving your mouse around the box that shows up (creates randomness), and a wallet pops up. However, in this situation you are extremely vulnerable because you are connected to the internet. Play around with the site a little, and get familiar with it, before continue reading below.
Use an old computer that you WILL NEVER connect to the Internet again to generate paper wallets. This will be your dedicated paper wallet printer. Keep it safe because you don't want someone malicious to have physical access to the computer. When storing larger amounts of coins, it may be worth it to destroy the computer after generating the wallets, as an extra precaution.
You need to download bitaddress.org's github repository. It is the link at the bottom right of their Web page. Now you will be able to run the website without being connected to the internet(it should never be connected to the internet again). It is a .zip file that contains the entire website, unzip the folder and run the html file.
If you enable BIP38, you get additional protection in the form of a password that encrypts your private key. That means that someone will need to physically access your paper wallet, as well as know your password, to steal your coins. I strongly suggest it. Once you have the program open, click the paper wallet tab, then check off the BIP38 option, type in a password(don't forget it!), and click generate.
Now that you've generated your BIP38 encrypted paper wallet, it's time to print it. You have to be careful at this stage because some printers, especially office printers, have built-in memory that stores print jobs. Someone could access this memory later and get your wallet. The BIP38 encryption protects you in this situation, because the printer doesn't know the password, but you still shouldn't do it. You might want to buy a dedicated printer just for this purpose or you can copy the private key and bitcoin address down with pen and paper.
If you want to keep a digital backup of your paper wallet then print it to pdf, but you shouldn't rely solely on the BIP38 encryption. Since it is a digital copy, it will be easier for malicious individuals to access the wallet, so it needs more security. Download and install TrueCrypt. Run it, click create volume, and then click create an encrypted file container. It will generate a file of any type and size that acts as a concealed and encrypted folder. Play around with it before using it for real. It is really powerful and can be useful for many other purposes. Then put the paper wallet's pdf into the encrypted container. Make sure you use a different password then you used for the BIP38 encryption. Then you can transfer the container to a usb drive and even upload it to a cloud service such as SpiderOak, Google Drive or Dropbox.
If you have an android phone or tablet, I suggest you install Mycelium. It is a powerful, light, and open source bitcoin wallet. Once you're in Mycelium, click keys, click the add key symbol on the top right of the screen, then click scan. Then scan your bitcoin address. Now you can keep track of the total balance on that address and because your private key isn't on your phone, there is no risk of your funds being stolen. When you decide you want to spend your coins, its as easy as clicking the menu button on the top right of the screen, clicking cold storage, scanning your private key, and entering your password (if you used BIP38).
If you don't have an android device, you can use blockchain.info to watch your bitcoin address from a web browser. Create an account with them. Then once you are signed in, click the import/export tab, and then click I understand. You'll be presented with a page that looks like this, allowing you to enter either your bitcoin address (if you just want to watch the balance) or your private key (if you want to move the funds).
The next best thing is to download Tails OS. It is a linux based privacy focused OS that you run from a usb drive. When you remove the USB drive, all traces of your computer session are erased. Install Tails to a usb drive, boot from the drive, and then go through the same process detailed above.
You can buy the Piper Wallet Printer, which is an all-in-one solution for paper wallet generation. Essentially it's a purpose built computer/printer combo that never connects to the Internet and comes preloaded with everything you need. You just press a button and a new wallet is generated and printed. It's $199 and you can buy it with bitcoin. I am not affiliated with them in any way.
Disclaimer: This post is intended solely to provide information. As I have no knowledge of individual circumstances and technical level, readers are expected to complete their own due diligence before proceeding with anything mentioned in this article. The topics discussed in this post are advanced and readers proceed at their own risk.