Before you delete anything just make sure that you have copied / saved everything in the way of files, folders, photos, etc that you need from the old computer. The best way is to save all your stuff to an external hard drive. Also make sure that you have all the discs for the installed programs so that you can use these on your new computer. If the new computer has a different operating system to your old one it could be that some of your old programs won't work. In that case you'll have to get the latest version off the internet.
Having done this, I would run the new computer for a week or so and you might find there's something that you're missing. If you've still got the old one this won't be too much of a problem. If, on the other hand, you've wiped the old one clean then it will be a problem!
Assuming that you've still got the original disc for whatever version of Windows your old PC is running, the best way to clean all the old stuff off the old PC would be to do a clean reinstall of Windows. This will automatically wipe everything off your old machine and you'll just be left with a clean, good as new "old" computer. (You may be astonished at how fast it is now running - and wonder why you splashed out on a new machine!)