I don't have a Sachs and I really know nothing about them, but I can try to explain some of the general stuff.
The carb's function is to mix fuel and air together into the right mixture so that your engine will run exactly right. If you have too much fuel in your fuel/air mixture, that's called a rich mixture, and it will cause your moped to run slowly and sound weird (four stroking). If you have too much air in your fuel/air mixture, that's called a lean mixture. If your mixture gets too lean, there won't be enough fuel/oil to cool the engine, and the engine can seize. Sometimes your moped will bog or run crappy if you're lean. (Past experience has shown that I'm pretty much incapable of telling the difference between a lean bog and four stroking... so if someone else explains that differently, listen to them.) When you're at full or mostly full throttle, the amount of fuel that the carburetor is mixing with the air is determined by the size of the hole in your main jet. If you go to a fix it day, you should look at some different size jets side by side. It always amazes me how such a tiny hole and tiny (barely visible to the naked eye tiny) differences in the sizes of the holes can make such a huge difference in how a moped performs. Generally speaking, you'll go faster if you have a bigger carb because a bigger carb will let you pump more fuel and more air into your engine. However, some of that also depends on how big your intake (connects carb to cylinder) is and how big the intake port in your cylinder is. Since I don't know anything about those two variables on a Sachs, I'll let the Sachs people tell you if that's a good performance investment or not. The same kind of thing applies to a high flow air filter -- it will let more air into your carb. If you put in a bigger main jet to allow more fuel (to match the greater amount of air), then that can help you go faster too. But again, I'll let someone else tell you if that's a good performance investment on a Sachs.
On the gearing, I'm assuming that "gear the rear sprocket down" means put on a rear sprocket with fewer teeth. A smaller (fewer teeth) rear sprocket will increase your top speed, but you'll lose some acceleration power. I would try to explain it, but it confuses me every time... so usually I just go read this a few times until I understand it again.//www.mopedarmy.com/wiki/Gearing
Racing tires wouldn't be very high on my list of performance upgrades. However, if it still has the original tires, or if the tires are looking cracked or worn, you should replace them. You might not go faster, but you'll be safer. Other little things you can do: always make sure your tires are properly inflated, clean and repack the wheel bearings, make sure your brakes and wheels aren't rubbing anywhere, double check chain tension, and make sure that you don't have air leaks anywhere.
I could link you to a bunch of articles and videos and stuff, but I know that if you go to a fix-it day, someone will explain/show all of this and more to you, and it will make a million times more sense a thousand times faster than if I try to explain it here.