I'm not certain which piece you're considering taking apart, but a derailluer generally doesn't need to be taken apart, and you can diagnose a problem without disassembling it, and the shifters -- you don't want to take them apart unless you're 100% sure they are the problem.
Don't disassemble the shifters until you've already checked everything else at least two or three times. If this is a road bike with the shifters integrated into the brake levers, there are a lot of little parts inside them and it is tough to get everything back together. If the shifter is the problem, most of the time it will be evident in if you operate the shifter and the cable does not respond as you would expect it to (shifting one way pulls more cable into the shifter, and shifting the other way releases some cable out of the shifter)
The order to check stuff is:
1 - Limit screws (you can just check visually and move the derailluer by hand
2 - Cable tension. You can redo this pretty easily
3 - Check cables/housing. Take cable out of the derailler and see if it moves easily through the housing. If not, replace one or the other or both.
You could do the shifter test in the first paragraph first or second.. If they clearly are not working, then you'll want to consider trying to rebuild them or just finding working shifters.