For the bitcoin wallet, your options are listed here:
http://bitcoin.org/en/choose-your-wallet with the most basic being
http://bitcoin.org/en/downloadfor dogecoin, your safe options are the respective windows and mac versions of the dogecoin qt client.
https://github.com/dogecoin/dogecoin/releases/download/v1.3/dogecoin-qt-v13-Win.zip -- Win
https://github.com/dogecoin/dogecoin/releases/download/v1.3/dogecoin-qt-v13-Mac.zip -- Mac
tl;dr each coin requires its own wallet program/client, and please stay away from online wallets, as quite a few of us just lost alot of doge with the whole dogewallet.com hack/scam.
Even though it takes quite some time for the dogecoinQT client to sync with the blockchain, please have the patience to avoid being screwed by online wallet services.
In the case of dogewallet.com many miners lost hundreds of thousands or even millions, and I myself lost 50,000 doge.