There are several ways to obtain Dogecoins (DOGE):
Mining for DOGE
Using your computer and some free programs from the internet, you can use your computer's processing capacity to mine for DOGE. It requires an amount of your computer's processing power and electricity to run your computer.
The programs connect to a pool of other miners who are all trying to solve blocks of equations.
"...which basically means having a computer program solve complex mathematical equations. Any computer can participate, but the more powerful it is, the faster it will go (and the more coins you will get!)."
- reddit Dogecoin wiki
Getting Started with DOGE mining
The basic steps are:
Get your wallet from
http://dogecoin.com/.
Join a mining pool.
Follow a mining guide for your preferred way of mining
CPU
GPU (faster!)
Receive the Doge and have fun!
Buying DOGE with traditional currency ($)
CoinTouch - Find Friends that Trade Bitcoin
Trading other cryptocurrencies for DOGE (BTC or LTC)
source:
http://www.reddit.com/r/dogecoin/wiki/index