Windows 7 comes with a "
Windows Easy Transfer" feature that is suppose to help you with that;
however, it requires a special USB cable you have to buy, and since you need to connect that cable between the two computers, it will be a hassle if you have the two computers in different places (eg. one in a room upstairs and another one in the basement, for example). Finally, even with all that, I don't think it actually transfer entire applications; those you still have to reinstall.
So let's forget about that. Then it depends on how many files you have to transfer and how big, but generally speaking, it's really recommended that you go buy a flash drive, as unfortunately there really aren't any other alternatives that are particularly easy to set up and use. If it's just a limited number of small files, you could use the Internet (upload or attach to email, then download), but that will be a slower way to go especially if you need to transfer large files.
Also note again that you will likely need to reinstall any programs you have on the old computer onto the new computer; merely copying the files will likely not work.
As an alternative to buying a flashdrive, you could always use a few CDs or DVDs (especially if you already have rewritables).