Electricity can be tricky. Some objects need wires run directly to them. Others simply need to be within the radius of a power radiator. I don't mean a power source, like a windmill or fusion generator. You can find the radiators in the same section as the conduits, and they have to attached to a conduit of some kind. The vanilla radiators have a ridiculously small radius. I modded the game to increase it 10 times the area. There are mods on the Nexus that do likewise.
With items like spotlights or laser turrets that need electricity directly run to them, you have to start with a generator of some kind. You can make windmills with no Science, so that is what you will probably start with. If I recall, you select the generator and press space. That should give you a wire that you can attack to an object. If it is close enough you can run the wire directly to your device. Most likely you will have to create a series of connectors to move power around your settlement. You run a wire from one connector to another in a daisy chain, until you get to your end devices.
There are all kinds of connectors. Most are in the Connectors and Switches sub-section of Power. The main kind is the Power Conduit. It is just a little metal plug that you can run a wire to and from. There are also pylons, which are good for going over long distances outdoors.
There is also a conduit sub-section has a lot more as well. This is not to be confused with the Power Conduits. It is all pipes that power runs through. I like the conduits. Start with a Conduit Junction (it acts like a Power Conduit), and then attach a line of conduit pipes from it. I put power radiators at regular intervals along the pipe. Typically I will run a line of conduit down the lengths of two walls of a building. With the modded power radiators, that will power all the lights inside.
Here is an example, with power coming from a windmill on the left, and running down into two lines of conduitI have read that Power Pylons also give of a radius of power, like power radiators do. But I have not really tried doing much with them.