I can tell you some things you've been doing wrong:
1st. You didn't read the instructions
on how to use the forum.
2nd. You didn't put your code in code tags.
3rd. You didn't tell us how you connected the buttons to your Arduino.
4th. You didn't tell us what makes you think something is wrong.
5th. (perhaps) You didn't use the option to enable pull up of button pins (see 3rd.)
6th. You are looking for a signal to get HIGH.Do you know for sure it is getting LOW at any time ?
7th. (big one !) You are setting buttonState to each button, and the last one you read will be left after reading them all.
Then you are either switching all LEDs on or off.
After that you are switching each separate LED on if buttonState is HIGH, and off again after the next LED is to be switched on (and it will).
This all happens in less than 1/1000 of a second or so (moderate guess).
All LEDs will be lit slightly, certainly not at full brightness to your eyes, if you can see any light emitted.
Make sure you understand how fast things happens in Arduino world.
buttonState is there to help you see some button has been pressed for some time, and to help prevent registering false button pushes.
Read about debounce again, more carefully, you left out something.
Your code can be optimized big time by counting a bit.
And using calculations you can also register multiple button presses if you like (but your sketch doesn't show you want that at this moment).
Do not try to solve all these in one go.
Take it step by step.