computer on, monitor no signal

I have an older E-machine with windows XP, that quit working several months ago.  Since I am a poor student and had much of my school work on this computer, I took the computer to a computer repair show, who said I needed a new mother board.  Relunctantly, I replaced it.  Now several months later, my monitor stater turning off and eventually wouldn't turn on at all.  I occasionally could get it to turn off if I pressed F8, but now nothing works.  The computer is running, and another monitor won't work either. 
Took it back to the shop, who were suppose to diagnose the problems, but they just repaired it saying it was the power source.  Now a week later, it is doing the same thing as before, but I can't get monitor to work.  Repair shop said computer is fried and there is nothing I can do. 

I don't trust these guys, I issued a complaint against them, but I still can't access my computer.  Is there anything I can do to get the monitor on so I can get to some of my files?  Or how I transfer everything to my laptop or a new computer?

I have no experience working on computers, but am willing to try.

I appreciate any help.

...Or how I transfer everything to my laptop or a new computer?...

Remove hard drive and connect it to new computer with this:

You can get this from Tiger Direct.....

http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=2329300&CatId=3770



They cost about $20.00