App for convenient media consumption
There is an App called Meteoric, which does exactly what you want. You can test the free version, which only supports 2MB downloads. And if you like it, you can buy the full version, which is with $1.99 quite inexpensive.

Here is a nice article with screenshots if you are interested (http://www.makeuseof.com/tag/download-file-ios-device-meteoric/)