As long as your engine's broken in, you can start using synthetic. It'll take a full quart if you take out the oil screen, the oil filter and that bronze-looking drain bolt at the rear of the crankcase on the left side... be prepared to clean up a mess if you take out the bronze drain bolt, there's a plate bolted right beneath it that catches the oil and spreads it everywhere. You don't really have to pull that bronze bolt, nearly all the oil will drain out through the oil screen hole anyway.
10W-40 should be fine... in fact, running a slightly thicker oil than the manufacturer recommends has long been known as a great way of extending engine life in severe duty environments... and running at WOT like our scooters do is pretty severe duty.
That oil will work fine in your scooter engine. The only thing I'd take issue with is that it says it's wet-clutch compatible... meaning it's missing some of the friction modifiers that can make oil really slick. It'll still work, though. I use Royal Purple 10W-40 Fully Synthetic Oil with Synerlec and add 8.5 grams of tungsten disulfide (WS2). Others use RedLine, Motul, or whatever they can find cheap... the point is to change it often. I change mine every 600 miles.
I do the engine oil, engine oil filter and gear oil all at the same time, since I'm there and working on the bike anyway.
After you've drained the old oil and put in the new, put your dipstick back in and run the engine for a couple minutes, snapping the throttle a few times. That'll flush oil through the system and purge your oil filter of any air. Then shut down the engine and check your oil level. You'll likely need to add a bit more to bring it up to the mark. I always fill to the high mark on the dipstick.
Remember, when checking your oil level, you don't screw in the dipstick. Just unscrew it, pull it out, wipe it off, then set it back in the hole without screwing it in, then pull it out to look at the level. Then screw it back in to seal it up, and have fun riding.
As for the gear oil, pull the drain plug (the one with the metal washer on it), let it drain out while putting a box or book or something under the front wheel to tilt the bike back a bit. Rotate the rear wheel a few times to squeegee off the oil from between the gears. Then put the drain plug back in and fill it with about 180 to 200 ml of gear oil. The manual says to use 75W-90, I use 75W-140 because it's thicker and will prevent gear wear. When the bike was new, every time I used 75W-90 there was a lot of "glitter" in the gear oil, but using 75W-140 put a stop to that.
The easiest way to measure the gear oil is to go buy one of those 2-cycle oil mixing giant syringes from Walmart. Then put in 4 shots of 50 ml each. Put the plastic hose that comes with the syringe over the oil bottle tip, jam it down hard. Then put the syringe in the other end of the hose. Invert the bottle and pull out on the syringe to draw oil into it to 50 ml. Then turn the oil bottle right side up, and pull the syringe off the plastic hose. The suction drawn in the oil bottle will suck the oil in the hose back into the bottle. Then insert the syringe tip into the fill hole of the gear casing, and squirt it in. No drips.
I just put in new taller rear gears (my Yager hits 86 MPH now. WOOT!), but after the gears are worn-in, I'll be adding 1.75 grams of tungsten sulfide to the gear oil. That coats the metal surfaces, reduces friction and wear, and helps to quiet gear noise.
