There are multiple ways to acquire Monero: You can mine it, you can exchange services or goods for it, or you can convert other cryptocurrencies and fiat money to XMR. If you choose the latter, the most convenient way to do it is to use an exchange.
There are multiple exchanges supporting Monero. Some are centralized, which usually have great liquidity and fast service, but requires you to provide personal information before starting the trade (KYC). Some are decentralized and don't require any identification, but usually have less volume and can be harder to use. There are also services that allow people to meet and perform a trade without third parties involved.
An incomplete list of Exchanges that support Monero is on our Merchants page.
Download the paper wallet generator at: moneroaddress.org and copy it to a USB stick (Direct link: https://github.com/moneromooo-monero/monero-wallet-generator/archive/master.zip).
Unzip and open the paper wallet generator (monero-wallet-generator.html) into a web browser on an air-gapped (airgap) computer that hasn't been used before, or has had a clean installation of the operative system.
Your paper wallet will have four important items:
Monero mnemonic seed: The mnemonic seed is a method of storing the entire wallet that is easily recognizable to humans. This is all you need to restore your wallet at a later date.
Monero spend key: The private spend key is used to send funds from the wallet.
Monero view key: The private view key is to view transactions entering the wallet. Commonly this is used to setup a view-only wallet which can see incoming transactions live on the blockchain as they are sent to a cold storage wallet.
At this point you have many options. You can print the wallet on paper, save it as a PDF or text on a USB stick, burn it to CD/DVD, etc. Most likely you will want at least two or three copies, stored securely in different locations. If storing digitally, encrypt everything with a strong password. If storing on paper, do not show the wallet to anyone else who can memorize your 25 word key, or take a picture of the wallet without your permission. Sending someone a picture of the wallet is the same as giving away all of your funds.
Whichever method you chose, be sure there's no copy of the Monero wallet left over on the device you used. You may need to securely delete the Monero wallet if you saved it to a disk, or make sure your printer does not save a copy in memory.
If you lose access to your Monero paper wallet the Monero will not be available to you or anyone else, ever. You wont be able to recover them!
Option to encrypt an XMR mnemonic seed: https://xmr.llcoins.net/
Download the html page and place it on your airgapped computer. Check the part "Encrypt/Decrypt Mnemonic Seed" and make sure you use "CN Add" with a decent password. Thanks manicminer5.
Now that you have everything you need, you are ready to send your XMR to your paper wallet. Simply send the coins to the wallet address you noted earlier. Make sure the address is correct, even if you copy-pasted it! Remember that if you send the coins to a wrong address, there is no way to have them back!
Because the Monero blockchain is private and untraceable, you won't be able to lookup your Monero Public Address and confirm that the funds have arrived like you might with Bitcoin. This is good for privacy, but bad for convenience.
To securely verify the funds have arrived at your wallet, you will need to setup a View Only wallet. This is where that view-key comes in. To create a view-only wallet see the entry: View Only Wallets
To verify the funds are still in your wallet and have not been spent you need to create a Cold Wallet with your mnemonic key (all your funds) on an airgapped computer with an up-to-date copy of the Monero Blockchain. When finished you will have to securely erase the wallet or connect it to the internet and it becomes a Hot Wallet.