If you want to buy EOS using Fiat money, the easiest way to do it would be through https://www.coinbase.com/ as your first step, here's how:
1 - Open a Coinbase account and link it to your debit or credit card (or alternatively, transfer USD, EUR or GBP to it)
2 - Buy BTC or ETH through your Coinbase account
3 - Send the BTC or ETH to an exchange such as https://www.kraken.com/ to trade for EOS, or alternatively, use services such as https://shapeshift.io/ or https://changelly.com/ (if you prefer to stay away from exchanges)
4 - Once you buy your EOS tokens, make sure you send them to your wallet (this mainly applies if you're going to use an exchange) and DO NOT leave them on any exchange. Buy a hardware wallet such as https://trezor.io/, https://www.ledgerwallet.com/ or https://www.keepkey.com/. Trezor is my favourite hardware wallet and it supports https://www.myetherwallet.com/, which means you can store any ERC-20 token on it, including EOS.
5 - After you receive your EOS tokens into your wallet, make sure that you register them by mapping your PUBLIC Ethereum address to the EOS PUBLIC address. Here's an article which explains how this is done for the EXODUS wallet, just replace anything related to EXODUS with the wallet that you will be using: https://goo.gl/36yDtS
Other helpful articles:
How to transfer EOS ERC20 tokens using MyEtherWallet: https://goo.gl/dgCsyJ
How to update the EOS public key mapped/registered to your ETH Address with MyEtherWallet (MEW): https://goo.gl/TSq52F
RE: The Dawn of EOS.IO