Hi and welcome to this forum.
Being pretty new to Arduino world a robot is already a little challenge.
You will be faced with hardware and software issues. But - as you say you are quite enthousiastic about Arduinos - go ahead!
My first remarks to your project:
1. 2 motors might be too less if you want your robot really be responsive. 2 are ok, 4 are better.
2. For your motors you will need motor shields as the Arduino itself is not able to deliver enough power (the current outputs are limited; see this link.
3. You will need at least two collision avoidance sensors (front and rear); this could be either ultrasonics or infrareds)
Do I need another component to power up everything (motor, leds, buzzers)?
Yes: at least motor shield(s); depending on the number and power of leds, you might need a kind of booster (in the easiest approach normal transistor(s) plus base resistor(s) will do); afaik the buzzer doesn't draw that much current, so the Arduino might be powerful enough to drive the buzzer - but look at the datasheet, to make sure you don't stress the Arduino's output.
Is an arduino uno sufficient or should I buy right now an arduino mega?
It depends how many pins you will need, so add all needed I/O and compare how much pins you got left (starting with the UNO; take into account that you can use almost all analogue pins as digital I/Os if the existing purely digital I/Os are not sufficient; if you need more pins -> either the mega or an additionla multiplexer chip for the UNO is needed).
Do I plug all this to the arduino board or to the breadboard?
I assume you are looking for a compact design, so you might go with a suitable motor shield which could be stacked on the UNO or Mega, but it should give you easy access to the remaining/unused pins to hook the sensors, buzzers and LEDs. If you have to go with transistors to power any LED and/or buzzers (maybe later a sound module etc.) than you will also need an external little PCB board to host those boosters.
If i have the hardware and with some correct coding, could this robot make lights (leds) or am I planning to do something impossible?
This is definitely possible with an Arduino platform - if you search with the right keywords you will find a lot of similar projects here in this forum or somewhere in the I-Net world. This will give you some more ideas for your project and you can benefit from others' experiences.
But my last advice: Think BIG, but start SMALL.
That means: Try to set up a master plan with your ideal robot, but then start with some small steps to get to learn how the things in the Arduino world fit together and then, when each single step works, merge them together as a whole project.