1. What kind of oil are you using in the transmission? Use FORD AUTOMATIC TRANSMISSION FLUID (Type F at most places). Going up hills on my Tomos normally means that if you DONT hit it at full speed... 15 is as fast as you're going to climb it. This is because if you're going full throttle at 15mph uphill the stress on the transmission wont allow the 2nd gear clutch to engage until you relieve some of the stress.
2. Clutches are cheap... labor is expensive.
3. You need a certain amount of skill when dealing with a moped transmission. There are manuals on www.mopedriders.org (or .com). You may need some special tools to get the job done correctly and safely.
Call up a dealer. Have him look at the ped. First and foremost... Change the transmission fluid and see how it goes.