scooter is probably not the most specific designation to be asking about.. The line between what requires licensing is generally based on engine power, typically above and below 50cc (and “scooter” engine power can range far above that).
True for most states: things at & above 50cc require special training and/or licensing, things below don’t. Though even with <50cc, getting training on how to handle yourself in traffic is always a Good Thing (someone mentioned MSF, which I’ve done and recommend as well). 50cc also tends to be the line drawn for what you’re allowed to take on to highways. In most cases, under-powered scooters are not allowed on highways over concerns of causing obstructions to higher-speed traffic.
For protective gear, most states (if not all?) require helmets.. but not much else. Don’t skimp on this purchase—it’s your best insurance against bad days! However, if you intend to drive in heavily trafficked areas (like cities or highways), you should also buy a jacket to protect yourself. I’d pass on the cordura/nylon stuff; they may do well against weather and are more “breathable”, but anyone who’s ever been “down” knows that the material holds up for about two seconds against asphalt, before it grinds/melts away to bare skin. Stick with leather, and not the soft, plushy kind.
Brands – eh. I’m a motorcycle-type person, but I’ve seen a lot of new-ish brands that look pretty or import styles from another country. They look nice, but just keep in mind that all mechanical things break down, and the more common they are, the cheaper they are to service and fix, the less money they cost initially and over time.
Insurance laws vary state to state. 15 years ago in FL you didn’t need insurance for a <50cc scooter. I think you might need it in San Francisco, but I’m not really sure. If I took a guess, I’d say it’s probably usually based on engine displacement (like licensing).