As Marcus suggested, for a taste, try Moodlecloud. This way you can get a feel for Moodle. For AV I have been using .mp4 (and .mp3 files, for audio), and that has been successful across platforms, as well as directly linking to Kahn Academy/MathBFF/others on YouTube. I am sure others would say there are better formats and techniques, but they work for me. Screen sharing though, mmmm collaborative file sharing on Google Apps, has been trialled, but I can't claim successfully with some students. You might want to try Office365 as a workaround here, but then I am likely not getting the real intent of what you would call "screensharing" Or would BigBlueButton be what you are looking for?
Moodle will help if you can provide your own servers, which does cost, but there are Moodle Partners, which also cost, but these costs can be reasonable, with some negotiation and clever management. There are some cheaper providers, but you are not really likely to get the service that you may need/want, you might, but then again.... You get what you pay for, really.