To upload new files (such as images and videos) to your media library, navigate to Media in your side bar and select Add New. This will take you to the Multi-file Uploader. You can upload files simply by dropping them in the designated box, or by clicking on Select Files, which will show a drop down menu where you can choose your files.
If your browser is having trouble using the Multi-file Uploader, there is an option to use an older version of the uploader that is easier on browsers.
When you drop a file into the uploader, a progress bar will come up. When it finishes, you can click Edit, which will bring you to the media editor; from there, you may make modifications as you wish.