Here's some first hand experience for you.
About a year ago I bought a 50cc moped to travel to and from work, started off wearing a heavy leather jacket, helmet and gloves. Bought a proper motorcycle jacket after a while but wearing shorts to work was just easier. All was going well, $8 or less to fill up, zips through the traffic and a lot of fun to drive up and down Great South Road.
But then one drizzly grey Sunday on the way to a morning shift, I didn't see a pedestrian crossing the road and he didn't see me either. He gave me quite a fright so I applied the breaks a little harder than I should have and I locked up and slid off. I didn't realise I hit my head and picked up my bike. Next thing I know I somehow made it to work with no recollection of the whole ride to work. When I went to sign in I was in no fit state to stand, let alone work. So they rushed me down to EC and I couldn't even fill out and ACC form. I spent the next 7.5hrs in emergency care and can only remember small parts and the last hour of my stay, thankfully cause it was very painful to even move. Not a pleasant way to spend a Sunday and the bike got scratched as and had steering slack.
I got away with a concussion and some pretty small patches of road rash which had to be bandaged for more than a fortnight and attended to for much longer. I now get headaches almost daily and have scars that will probably never go away. Not to mention a $140 pair of leather chucks I completely ruined.
My point is, accidents happen far more often on 2 wheels than they do on 4 and when they do happen they WILL be far worse. A moped is fun and cheap for a while but sooner or later you will get a car for various reasons. Just be very safe and do anything you can to keep yourself protected.