There isn't any way to copy programs themselves onto your system, as they all need to be reinstalled on the new hardware. You CAN however keep ahold of your data.
There are ways to go about this. (CD burner is best, but I'm guessing you don't have one in the old system)
As mentioned above, you just put the old harddrive in the new system, but here's how to do that in a little more detail. First get the new system up and running with Windows installed. Now shut the system down and open up the case. Find your harddrive and on the back there are is going to be a jumper block. On the top of the drive there *should* be a picture showing the 3 (or 4) different possibilities to select: Single (only drive on channel), Master, Slave, and Cable Select. Cable Select is of no importance to us. On your new harddrive in the system this should be set to Single (most likely). You need to pull the jumper off and put it to master. Now take the hard drive out of your old computer and set its jumper to slave and put it into your system. You will need to hook it up to the free connector on the same ribbon cable as it hooked to the hard drive already in the system. (Be sure to get pin 1 lined up correctly, just match the other drive or look for the colored wire - that's pin 1). The only other connection on the hard drive is its power connector. These only go in one way. Once you get it all hooked up just start up the system and when you get into Windows you will find the old drive listed under Windows Explorer (file manager).
Now you can copy and paste all you want.
Once you are done, remove the old harddrive from the system and make sure you change the jumper back to "single".