Parking safety shouldn't be a problem as most bikes can lock the wheel in place.
When the moped is off, turn the wheel to the left while turning the key in the opposite direction of starting it. This should lock the wheel in a left turning position. People won't be able to simple roll it away and it's too heavy to pick up. Of course that doesn't mean people can simple toss it in the back of a pick up truck and drive off with it. But that's unlikely ever going to happen as it's simply not worth all the effort for a moped.
Weather to avoid would be rain and snow. I've driven in heavy rain before and it didn't give me any problems. But it's hard to see if you're wearing a helmet. And those rain drops feel like needles poking you! It's hurts! lol
Laws may differ from state to state, but in most states, you are REQUIRED to drive and follow the some rules and laws as cars.
So you must drive in the center of your lane.
Driving on the shoulder is illegal unless you're going way under the speed limit and thus causing traffic behind you. In which case you may move to the shoulder lane to let cars pass you.
Driving on the highway is, in most states, illegal if your moped is under 150cc. But the highway is required to have signs stating that.
So if there's no sign, no problem driving there.
But beyond that,driving a moped isn't much different than driving a car( legally speaking ).
Now in regards to tips:
Always wear a helmet. duh
Try to wear long pants and shirts. This'll ensure you won't get road burn in the event of an accident.
BUT make sure you clothing is as tight to you skin as possible. Anything too baggy can act like a parachute and slow you down when driving.
Follow traffic laws as any car would.
When stopping at a red light( or in general ) place BOTH feet on the ground.
Take care of your moped. Check the oil and such. You take care of your moped, your moped will take care of you.
Remember to turn OFF your blinker! I can't tell you how many times I forgot to turn that damn thing off and just kept on driving like an idiot... lol Unlike cars, with bikes you have to manually turn them off.
That's about it.
Oh and last but not least, don't be a dick. Be respectful to the drivers around you.
And if you see another biker driving by you and he/she waves at you, return the wave. If not, don't.
The biker wave is a way to acknowledge other bikers. However, most bickers don't consider mopeds and dirt bikes to be TRUE bikers.
But if they wave at you first, take some pride and wave back.
Edit: So many typos! <_>