I want to add some new functionality on the administration product edit page; providing the ability to upload videos and audio.
How can I achieve this?
you can upload whatever from there if you remove the restrictions of file types from uploader. Little grep displays how it is done currently in core code:
grep 'setAllowedExtensions' app/code -rsn
app/code/core/Mage/Adminhtml/controllers/Catalog/Product/GalleryController.php:40: $uploader->setAllowedExtensions(array('jpg','jpeg','gif','png'));
app/code/core/Mage/Adminhtml/Model/System/Config/Backend/File.php:62: $uploader->setAllowedExtensions($this->_getAllowedExtensions());
app/code/core/Mage/Catalog/Model/Category/Attribute/Backend/Image.php:58: $uploader->setAllowedExtensions(array('jpg','jpeg','gif','png'));
app/code/core/Mage/Catalog/Model/Resource/Product/Attribute/Backend/Image.php:57: $uploader->setAllowedExtensions(array('jpg', 'jpeg', 'gif', 'png'));
app/code/core/Mage/Cms/Model/Wysiwyg/Images/Storage.php:281: $uploader->setAllowedExtensions($allowed);
app/code/core/Mage/Dataflow/Model/Convert/Adapter/Http.php:78: $uploader->setAllowedExtensions(array('csv','xml'));
app/code/core/Mage/Dataflow/Model/Convert/Iterator/Http.php:49: $uploader->setAllowedExtensions(array('csv','xml'));
app/code/core/Mage/Dataflow/Model/Profile.php:150: $uploader->setAllowedExtensions(array('csv','xml'));
app/code/core/Mage/Dataflow/Model/Session/Adapter/Http.php:50: $uploader->setAllowedExtensions(array('csv','xml'));
app/code/core/Mage/ImportExport/Model/Import/Uploader.php:61: $this->setAllowedExtensions(array_keys($this->_allowedMimeTypes));
app/code/core/Mage/XmlConnect/Helper/Image.php:93: $uploader->setAllowedExtensions(array('jpg', 'jpeg', 'gif', 'png'));
rest is just implementing the file display in frontend with various plugins available on the internet. As your question is very abstract (try to ask better) this is the minimum that I can provide to answer it.
There are 2 ways to go:
Both ways are good, the best one depending on your requirements. No point in actually developing this feature yourself.