The best way to create a safe wallet and get your dogecoin address is to download dogecoin Multidoge Wallet from http://multidoge.org/?dl=win Here you can download the wallet for windows and install it. After installation run Multidoge Wallet and you will have your own wallet you can create as many address you want to receive dogecoins. Please note that on http://dogecoin.com/ you can get many versions for any operating system you have. Also if you are new don't install dogecoin core wallet as it uses allot of space and you have to wait to download whole blocks which are many gigabytes.
Other option are too there like online wallets but they are not so safe compare to your own wallet. There are some online wallet services where you can get your address.
To get Bitcoins you need a wallet and address with which you can receive or send bitcoins. So you don't need any bank account here, you just have to download the Bitcoin wallet for your operating system and after installing it. You can create Bitcoin Wallet.
Here is the link
You can also create online wallets. You must take care of its security. Because some browser plugins can get your wallet detail. So always use it on less or non-plugin browser. Or create Google Chrome another user where no extentions or plugins are installed.
Some Online wallets are.