The standard troubleshooting procedure for me is as follows:
1) Try to start moped. Pedal/push like hell.
If you can't do this you are in trouble. Your transmission is messed up or you have a seized engine. If you do this and everything seems fine but the bike isn't running, move to..
2) Take spark plug out. Is the plug covered with gas?
If yes, check for spark by placing the metal on the spark plug onto a ground like the cylinder. Pull in decompression/clutch cable and try to start. See if there is spark. If no spark, read fred's guide .
If no, clean your carburetor.
Those two steps usually fix 80% of problems. Of course, there could be many other problems. You could have blown a piston ring, your clutch could have came loose, you may have no compression due to a stupid little metal valve.