If you've bought a Honda, Suzuki, Yamaha, Vespa or other Japanese/Italian scooter from a reputable authorized dealer, you won't have any assembly to do. It will arrive fully built and tested and it will be ready for you to ride.
If you bought a mail order chinese scooter from an out of state importer, you''ll have some work to do. It will arrive in a steel cage inside a cardboard box.
First remove the cardboard box. This is the easy part! Then you'll need to take apart the steel cage by removing all the nuts and bolts holding it together. At some point in the process you may need a second person to help you by holding the scooter upright while you remove the last few bolts and possibly the bailing wire used to secure the scooter to the base of the cage. If you're lucky and your scooter has both wheels attached, you can roll it out of the case and set it on the center stand.
Now you can look for "assembly instructions", which with the average chinese scooter will be one sheet of paper which says something like "attach the rear view mirrors, install the battery, etc.". You may have to use some ingenuity to figure out how to do all this, but it shouldn't be beyond the capabilities of any amateur home mechanic. It may help if you have a decent socket set, screwdriver set and set of metric wrenches since the supplied "tool kit" usually leaves much to be desired.
Batteries are typically shipped "dry" with the acid in a separate container. You have to add the acid to the battery and leave it for 30 minutes of so before using it. After this it shouldn't need to be charged to start the scooter.
Once you have all the parts attached (and that will include at least mirrors, seat, rear carrier, battery and floor mats), and you've waited about 30 minutes after adding acid to the battery, you'll be ready to start the scooter.
If you just put fuel in the tank, you may have to crank it quite a few times before it starts since the fuel has to get to the carburetor and fill the float chamber before gas gets to the engine. You can prime the carb by disconnecting the fuel hose from the vacuum valve and connecting it directly to a small funnel and adding a little gas, then reconnecting the fuel line to the valve and adding fuel to the tank. However if you don't know what you're doing, just try to follow whatever instruction came with the scooter. Use premium gas. Most scooters seem to be setup to run on 91 octane fuel.
The final point of assembly is to check everything. Check the tire pressures (mine were 80psi and should have been 35psi), check that every bolt and nut you can see is tight, especially the ones holding the exhaust and muffler - and the wheels! Look around for lose wires or rubber hoses that don't seem to be attached to anything. It's a lot better to spend an extra 30 minutes checking for lose connections now than to ride 10 miles and break down and then have to find which part has fallen off or become disconnected. When you do take a test ride, take a cell phone and a tool kit with you, just in case you need to call for help or do your own roadside repair. I'd recommend riding around local streets so that you are in walking distance of home until you've got 10 or 20 miles on the scooter and you're reasonably confident that if anything is lose it will have made itself known.