Program searched: Record and convert videos?

Hello everybody,

for a school Project about New/Social Media ans Video Streaming sites like Yahoo in the net I would like to integrate some video material. Part of the presentation will also be to show a video online (from a laptop) and simultaneously “offline” on a smartphone. Thus I need a program to record videos from the Internet and convert them to be able to play it also offline on an iPhone for instance. So far I could not find anything that was easy (enough) to handle for me and that is able to record AND convert. Can anybody recomment a (legal) software program or anything else I could use?

Thanks in advance.