I recently bought Fallout 3 GOTY version. Before I owned the normal version of Fallout 3. Do i need to restart my game? If not how do i make the DLCs work?

The DLCs should come on a second disk, at least mine did. In that case, just put in the second disk and install the contents. What system are you on? Xbox, PS3, PC? I don't think any of those you have to restart. You could always backup your save games and try playing the GOTY on them.