How do I use a battery charger with a Piaggio 50 Scooter

Charge the battery with your battery charger from anywhere you can find a positive wire like on the starter or a light bulb and ground to any bolt.

Try Walmart- they list numerous batteries for Piaggo fly. what size is the scooter?

It sounds like your alternator is not charging the battery. Fit a new battery remember to undo negative lead first to remove and fit positive lead first to refit the battery. When the new battery is filled please charge it using less than 1 Amp DO NOT use a car charger or try to jump start it. Your new battery should show 12.5 to 13 volts on a meter for a 12 volt system and 6.3 to near 7 volts for a 6 volt system. Start the scooter and check across the battery terminals your meter should show about 14.4 V for 12 volt battery and about 7.5 V for a 6 Volt one. No voltage increase means your charging circuit is not working. Either the alternator or rectifier(regulator) are not operating correctly. Worst case scenario is both units dead.

Re-set the idle adjustment screw on carb? Also check fuel / air filter.