I strongly recommend visiting
http://www.oregonvintage.com/Scooter.html and at least downloading the GY6 engine manual, (right-hand column, about 3/4 of the way down the list). Even if it's the 50 cc model the general procedures still apply.
This manual contains very detailed step by step diagnostic procedures to deal with lots of problems.
These scooters are fitted with a relatively sophisticated CV carburettor blessed(?) with a couple of devices that are a constant source of problems: the automatic choke or "by-starter" and the ACV (air cut-off valve).
Before doing anything else, check to see what make the spark plug is. If it's an unknown Chinese brand like "TORCH", change it for an NGK C7HSA or CR7HSA and even if it doesn't solve the problem, it'll be an improvement, in starting, idling and general behaviour, also the Chinese one WILL give trouble sooner rather than later. If your funds run to it, fit an NGK Iridium C7HIX or CR7HIX. Expensive, but a worthwhile performance improvement.
It's a long shot, but the correct valve clearances are vital to ensure reliable starting and correct running. It shouldn't need them adjusted this early, but one of the characteristics of Chinese engines is that their valves tend to close up their clearance early in their life. I believe this is due to "valve seat recession", where the soft alloy in the heads causes the valve seats to sink in during the engine's early life and close-up the clearances.
There's a video here
where it details the valve adjustment procedure together with comments about how if affects cold starting.