You're right, it's almost certainly not worth repairing a 6 year old desktop, especially if you're not techie and can afford a new one without serious hardship (you can probably keep it running with a good internal clean and an operating system reinstall if you wanted to and no components are actually physically failing, but it'll never be a powerhouse of a machine...) A desktop would likely be better value for money if you are not bothered about the portability of a laptop.
Backups - just back up any documents you can't live without, pictures, letters, etc. I wouldn't bother with anything else, software etc; you can just redownload/reinstall off CD when you get the new one; starting completely afresh is probably good. Drag&drop is fine!
To record TV you'll need, as you correctly say, a tuner in the PC, which are not necessarily standard in a lot of cheaper desktops but you could always add one yourself or ask the place you're buying it from to do it for you (if somewhere that'll let you customise a bit); it's not hard. Go for one with a very large harddrive (at least 1TB, preferably 2) if you want to record TV, those files are enormous and only getting bigger now that HD quality is pretty standard.
If you're not a gamer you won't need to worry too much about the graphics card; I'd probably get something with at least a low end dedicated card rather than just integrated graphics to help with the TV viewing but it's really not a major issue.
RAM - refuse to even look at anything with less than 4GB, more is always better but again, you're not a gamer so there is a point of diminishing returns. I'd get something with 4, 6 or if a good deal 8GB.
Any PC you'll buy now will probably come with a free or dirt cheap upgrade to Windows 8 to redeem when that releases. I am personally extremely on the fence about Win8, but, y'know, take the free upgrade and see if you fancy installing it when it's out and people have written reviews of it. Win7 will be fine in any case. Other than that, software wise, decide if you fancy paying out for a MS Office license or if you're happy to use the free alternatives like OpenOffice. I personally haven't paid for an office suite in 8 years but some people worry about compatibility or very specific functions that might not be exactly the same - depends what kind of user you are. Do not pay for antivirus; free alternatives are fine.
Good places to look for deals are Ebuyer, Scan, Dabs and Microdirect. Do read the specs properly, not all prebuilt systems on those sites come with Windows installed which I assume you want! Hope that helps :)