How to fix my ceiling lights?
How can I figure out what's wrong with them and fix them myself?


Equip yourself with the basic tools needed. Make sure you understand the basic principles of household electricity. Make hypotheses, test those hypotheses. Act on confirmed hypotheses.

As far as I know, about 1 person a week is killed in the USA by accidental electrocution (and many times more in the construction industry). You seem a little inexperienced so I hope you'll understand that folk contributing to this website don't want to contribute to those statistics. I recommend you learn about electrical safety from someone with experience. Perhaps there are local evening courses you can attend or you know someone who is qualified and willing to teach? Fixing broken household electrical items is something that careful householders can do but it isn't something you should rush into in a state of unprepared ignorance. Think of it as akin to knowing nothing about cars but wanting to replace the brakes on the car your wife uses to take your kids to kindergarten.

Equipment wise, you might need a set of insulated screwdrivers and a category-II multimeter that can measure AC voltage.

You need to have a basic understanding of voltage and current, the purpose of fuses, circuit breakers and RCB/RCBO/GFCI devices. You should understand what live/hot, neutral and ground/earth wires are for and how they are typically connected in homes in your country or locale.

You need to understand exactly in what ways this investigation can kill you and how your attempts at repair might kill other people. You need to be really sure you know how to work safely and how to always have the system in a safe state.

Some hypotheses you could test:


The replacement bulbs are broken.
There's no electrical power reaching the bulb holder.
There's a fault in the switch associated with the light fitting.
The fuse or breaker in the main-panel has cut-off the circuit to isolate a fault.


Here's a few things you might do to test those hypotheses


Try swapping bulbs with ones in a room that has working lights.
Use a multimeter to carefully measure the voltage in the socket.
Turn off power at main panel, Remove the switch, examine it, swap with working switch.
Examine fuses and circuit-breakers for indications that they have blown or turned off. 


If a fuse has blown or a circuit breaker has turned off, you need to identify the cause before replacing or resetting.