WinX Video Converter is a free application that will solve those problems. It's very intuitive, it supports lots of formats, it works really well and it's totally free.
It supports the most common video formats and can convert from one to another without any kind of problem: avi, mpg, DivX, mp4, mov (QuickTime), MPEG1/2, vob, flv, wmv... you will be able to play your videos on any device.
You can even select only a part of the video and convert only that part. It's very easy.
If you are a novice let yourself be guided by the program. If you feel a bit more professional, you can control some advanced parameters such as codec, resolution, format, bitrate and more.