Advice please

[dma]

Hello,
I am looking to change my old analogue TV and video for new digital ones (and about time). I have seen a nice digital sony TV that has HD and freeview - so that is sorted. But my son tells me I will also need a dvd recorder as the my video player is no good. I've done some research and in order to record whilst watching another freeview channel I need a dvd recorder with built in freeview - is this correct?
But I also watch the Hotbird channels @13oE. Hence a need a dvd recorder that will record freeview (whilst I watch another freeview channel) and that can record my satellite channels (I currently have a digital sat. receiver but it is not a PVR type). I was thinking of buying one with a hard drive so that I don't need to use discs. I would also want the dvd to play divx and avi files.
So, what dvd recorder do I need/can do all of this? I'm on a tight budget of £200. Can I find something with freeview tuner that will also allow me to record my satellite channels.
Any advice will be welcomed.