Where to start?

[Quiksilver3133]

So I picked up a new (to me) 1986 Elite 80. I am versed in a couple tasks (i.e. oil change, spark plug, air filter), but I am by no means an expert in automotive mechanics...

This particular scoot was allegedly ridden last year and the owner completed an oil change and plug change before the fall. He basically just parked it once before it got cold and hadn't touched it until we rolled it onto the truck a couple weeks ago. It's got a new battery, full tank of gas.

Right now, it's getting power (lights come on), but the starter isn't working (well). It turns over like, once, if at all. Some additional whining noises. Nothing beyond that.

I'm not sure where to start here. There's no kick-start, and I know the starter is getting power, but its like the engine reacting to not being properly winterized. Anybody have a good idea on what to look at first?

[Johnniespeed]

I would first focus on getting the starter to always operate properly. That means being sure the battery is good, ( not all new batteries are good ) fully charged, and that the starter itself is healthy, ( spins the engine over quickly ) You will want the starting system to be bulletproof and reliable, so as not to strand you anywhere.
Once the engine spins over quickly, I would test for spark, then probably attempt to start the engine.
Looking into the fuel tank, you should see clean fuel and no rust. Don't be shocked if you have to clean out the carburetor, they are rather sensitive to dirty fuel.
Old tires are not the safest , you may wish to consider new tires for your own safety.
Hope this gets you pointed in the right direction.
Post some pictures and tell us of your progress.

[Kevinstonge]

To reinforce johnnie's recommendations, it sounds like low voltage at the battery to me. Trickle charge, and try again.
Most of the 80's didn't come with an inline fuel filter so it's not a bad idea to add one. Definitely clean the carb as most people do not store to scooter the correct way.
When you're ready to replace the tires, get yourself some tubeless 3.5x10's much safer and better balenced. I like continental zippys personally.