It looks like batteries are like fluids, and when you “fill” something they get shuffled about, and are just gone when you use them up…so you can’t recharge them.
That makes lighting a problem, so I’m curious how people solve this problem. In particular if there’s anyway to get stationary power/batteries/lighting going on? The building I’m in has a solar panel, but no lights, and doesn’t seem to have any way of connecting anything to power. I’d love to know more about power.
Unfortunately, the only way to have installed lighting in your shelter of choice by crafting it via the Vehicle Crafting Menu. I use swappable storage batteries that I charge outside in a solar charging rig I built thru (again) the vehicle crafting Menu.
Your lighting options are Headlights, flood lights, and overhead lights.
You can set up a mini generator system using a storage battery, small engine and an alternator, I quite like using the 0.3 litre ones you get out of lawn mowers for this. You can also use atomic lights and install those in a vehicle frame if you have/can make a few.
if you’re lucky enough, find the Robots book, it will allow you to craft the control laptop, after that head to a lab or any place with turrets, hack them and and dismantle the turrets, than put them back in your base or whatever needs light, as they provide huge lighting, enough to light up the whole evac shelter
Lighting on surface-level and higher buildings is pretty easy (with good tools). Demolish a window or a door of your building. Park a salvaged vehicle next to it or construct a wheel-less “vehicle” if you’re working on a second-story project. Weld on a series of Frames through the gap. Put a Door on the gap frame, aisles (or various vehicle work stations) on the other frames, and headlights or floodlights where you want light. Power the vehicle with enough wind and/or solar and you’ll be good to go for as long as you keep the base.
Underground light and power are harder. You’ll have to refuel whatever you’re using, whether it’s a fireplace, oil lamp, flashlight, fuel-powered “vehicle”, or “vehicle” running on batteries that you charge upstairs with solar and wind. Fireplaces work better now than they used to for this - as long as you have access to lots of wood, you can light fires that last all day without tending.
Atomic lamps and turrets are another viable option for permanent lighting, but hard to come by.
Hopefully at some point we’ll have the ability to connect power sources and power sinks directly to walls and roof tiles, but I’m definitely not up to coding it, so I’m not going to get on anybody else for not wanting to, either. 
If you get sick of it, you can always just build a bunch of wind turbines, drop them on the roof, throw a bunch of copper wire and light bulbs into a lake, and use the cheat menu to spawn overhead lights. 