Misko, I assume you're in the U.S.? Go into any grocery store; somewhere in the front should be a display with a bunch of sorts of phone cards, gift cards, etc. Find one of the Visa or Mastercard gift card things; they come in denominations of $25, $50, $100, and a variable-price one that you can fill with anything from $20-$500 most commonly; the cost is that plus a small (~$5) activation fee. You can use them online just like a credit card, but the only money that can be spent is what you put onto the card when you bought it. Typically there are no fees beyond $2 per month after a year of inactivity, but since they're so small, that's unlikely to ever affect you.
It's by far the easiest way to convert your cash into an eletronically-usable form, with the added bonus of being very safe; if you use a fake identity and shell email account when you register it, there is literally nothing to connect you to the card, and even if the card information is stolen, all you lose is whatever was on the card at the time, rather than having to deal with the hassle of deactivating a credit card and have charges cancelled. Obviously you aren't perfectly cost efficient because of the initial fee and because you usually end up with a few cents hanging around on it, but IMO it's well worth it.
Oh, and when you get a bank account, here's a word of advice: NEVER USE A DEBIT CARD IN AN ONLINE TRANSACTION. DON'T DO IT.