On a moped, gas starts in the tank, goes through the petcock, through the fuel line, into the carb through the fuel fitting (banjo), into the float bowl, out of the float bowl through the jet, then into your engine. If any one of those things is blocked, broken, or not working properly, your moped will not work because no gas will be getting to the engine.
If the guy said you needed a new petcock, you should probably just get one. They aren't all that expensive, they're pretty easy to put on, and the old ones seem prone to leaking/being clogged. Start with a new one, and that's one thing to scratch off your list. (This is also a good opportunity to clean your gas tank, if it's dirty. See here).
Anyway. Turn your petcock on. If fuel flows out of it in a pretty decent stream, move on to the next thing -- fuel line and fuel filter. Usually it's pretty apparent if these need to be replaced. If you don't have an in-line fuel filter, get one.
If you have fuel flow into the carb but no fuel flow to the engine, your carb is dirty. Clean it.
You'll know you're getting fuel to the engine if trying to start the moped makes the spark plug wet with gasoline (if you're testing this, try to start it a handful of times before checking the spark plug. If it's dry, try choking it, then check the spark plug.) If the spark plug is dry, then one part of your fuel system is still clogged or otherwise broken/malfunctioning. If the spark plug is wet and it still won't start, check for spark. Also read Fred's Guide.
