Buy BTC first from a place like http://coinbase.com then exchange it to DOGE on exchanges like http://cryptsy.com. Alternatively you can try http://reddit.com/r/dogemarket but it's riskier.

I started writing a guide covering how to do this since at the moment, it's quite a pain with no direct USD to doge exchange.

http://howtodogecoin.com/exchanges.html

You need a minimum of 4 accounts and can expect it to take 4-7 days your first time through. But so worth, to the moon!

If all you want is a few doge, set a computer to mine for the afternoon. Even on a relatively weak computer, you'll end up getting a couple hundred doge.

EDIT: I only do this as part of a pool. I don't know how it is for solo mining.

You can also buy on ebay...with a markup of course.