Yes, better batteries will do all 3. Lithium of the same nominal capacity will be much lighter giving you better acceleration.
Lithium's voltage under load will sag less (as long as you use lithium with the proper power rating), so the resulting higher voltage the motor sees will give you more speed. Get 23 or 24 cells in series if going with Lifepo4 (nominal voltage 3.2 or 3.3V per cell) or get 20 or 21 in series if going lithium polymer (3.6-3.7V nominal per cell) to ensure the higher voltage.
You will gain autonomy because lithium has little Peukert's Effect compared to lead batteries. Your lead batteries have a nominal capacity rating based on a 20 hours discharge rate, but you are discharging them well over 20 times that fast, and when you do that lead batteries can deliver far less capacity than their nominal rating. Add in that you can't very deeply discharge lead without prematurely killing them and you end up with 30-50% less real world capacity than the same rated capacity of lithium.
A cheap and easy upgrade is to use thicker power wires both from the battery to the controller and from the controller to the motor. This will result in less heat lost in the wiring for a bit better efficiency, and it also increases increases the voltage the motor sees for a bit better performance.
Anything you can do to improve aerodynamics will improve top speed and efficiency (range). Slight changes in riding position or a set mod that lowers the seat can pay surprising dividends. For aerodynamics think of your scooter as pulling around a trail of turbulence, so it's not only about pierces through the air easier in the front. It's actually more about letting the air flow smoothly and less turbulent away from the rear of you and the scooter. Simple things are:
1. If you have 2 kickstands, get rid of one.
2. If you have a rear rack you don't use, remove it, or even better get an aerodynamic backpack to use instead.
3. Instead of a rear rack or backpack, build an aerodynamic tailbox that not only will greatly enhance aerodynamics, but giving it a lid can give you a lot of extra storage capacity. Just don't go filling it with a bunch of useless weight to carry around and negate much of your lighter battery advantage. That kind of one-off mod will also make your scoot unique and less likely to be messed with or stolen.
The way you ride can greatly affect range, and electrics are so cheap to operate that riding hard becomes habit, but for those times when you need maximum range do small things like look ahead and coast to avoid using brakes. When you can use timing to legally avoid full stops then that avoids accelerating from a stop up to that speed, and low rpm acceleration or starting from a full stop is very low efficiency for the motor and the majority of the energy you draw from the battery turns into heat instead of work done moving you and the scoot.
Hills chew up energy, so often you'll find that a flatter longer route can use less energy. Get yourself a Cycle Analyst, which is an invaluable tool in tracking your energy usage to help gain a better understanding and feel for performance and consumption and many other useful things for anyone using a light EV.
