I'm no expert; what I say on the topic is just talk, not advice. If it was my kids, I'd turn into my Mom and say No. Possible. Way. Motorscooters are not safe. They cannot be made safe.
...But if you are gonna, here are some tips: Don't assume "cute" equals "harmless." A motorized two-wheeled vehicle is more dangerous to a youngster than a single-shot rifle, and just about as dangerous to bystanders. If you wouldn't hand a child or teen a .22, don't put them on a scooter.
Seriously consider MSF class for any would-be rider who qualifies for the course.
For a younger rider, maybe you should take the class -- and make copious notes.
The scooter itself? 50 cc or smaller. Automatic transmission. Modern-style, with a strong frame under the body components. Four-cycle engine, you'll be happier and the machine will last longer.
Manco imports pretty good Red Chinese machines and that's about as low as I'd go in quality. They come in under $1K new. Some Assembly (usually) Required, including some safety-critical items.
Better machines will last longer and offer better service support; look to your local scooter shop for the major brands. In Indianapolis, Speed City Scooters is the major dealer, from Vespa to Kymco. Genuine's Buddy 50 is a good example of this level. Yamaha and Honda also make very nice small scooters, both classic-looking and zoomy modern styles. If you buy used, look to the major brands first. No-name Chinese imports are cheaper but can be tricky to keep running and I would not buy one used.
Please remember that Junior and his kid sister believe themselves to be immortal. Good instruction, proper safety gear (helmet and gloves, minimum) and getting them to apply and use same is absolutely critical.
Riding can be a fun hobby and can build valuable skills. One of my coworkers rode a 125 cc motorcycle from the moment he had a license until he was able to buy a car himself, years later; his Dad figured he couldn't get into nearly as much trouble as he might in a car. He survived intact and is a skillful, careful driver. He also hasn't owned a motorcycle in decades. YMMV.
INDIANAPOLIS HAMFEST, 2015
1 month ago