I can suggest three ways.
The ideal way is a face-to-face cash transaction with someone local to you. Try
localbitcoins.com to find someone who can hook you up. Pros: Relative anonymity compared to other methods. Con: there may not be anyone near you willing to sell bitcoins. Plus you actually have to leave the house.
Failing that, I'd try
bitinstant.com. Once you create a transaction online, you can deposit your cash at almost any bank, plus certain stores like CVS and Walmart. Pros: Speed. Still somewhat anonymous, but not as anonymous as the localbitcoins method. Some places might ask some information from you when you make the deposit, but as far as I know you're perfectly free to lie your ass off with no repercussions. Cons: You still have to get up off your ass and GO somewhere to make it happen.
And finally, you can try an exchange like
CampBX.com (which I have not personally used) or
Mtgox.com (which I have). Pros: You can do it from your computer without having to put on shoes. Or underwear. Cons: With mtgox, your account (and thus the amount of money you can move in and out of it) is limited until you provide information required by the government. So much for anonymity.
No matter how you get your coins, I strongly suggest you NOT store them in any sort of online account. Storing them on your own personal computer is a better option. But storing them in a "paper wallet" is the best option.
Basically you send your bitcoins to an address that you create, but that is NOT tied to your computer or an online account. The key to that account (think of it as a password) exits only on a piece of paper that you print out and keep.
