Are you getting a spark? To find out, take out the spark plug and leave it attached to the wire, meanwhile hold in the clutch handle and have someone pedal so the engine tries to turn over while you hold the spark plug close to engine crank case and look for a spark. If you're not getting a spark, there could be any number of things wrong, but check the following:
- new spark plug?
- wiring (which is always a big if, since there seem to be ungodly amounts of wires on these mopeds
- rear tailight/brakelight bulb burned out? This was part of the reason that my Sachs was not running when I bought it. The brakelight served as a connection somehow for the rest of the electrical system. (Interestingly, Volvo uses a similar feature on their older cars...you can't shift into drive or reverse if a brakelight is out.)
I have the manual for the Sachs 504/1 A engine. If this is your engine, e-mail me and I'll send it to you .pdf. I think that you will have that engine.
I was in the same boat as you a month ago...even worse off actually since I didn't know anything about combustion engines, much less mopeds. But, it's true that while they are quirky, they also tend to start right up just as long as you have gas getting to the cylinder and a spark to ignite it. It's quite a thrill when you get it started the first time and (hopefully) are able to take it for a spin. Never did 27 miles per hour feel so good.
Best advice though would be that if the bike looks like it's in pretty good shape, don't go tearing it all apart...it will probably start with just a little work on the carb, spark plug, or electrical system. I say this because while this site is by far the best resource, it also tends to make a new moped owner question whether everything is wrong. So rather than suspecting once thing after another, keep the diagnosis simple at first.