Buttons to Binary display and Hex Display for teaching

Good afternoon, have an idea for a display for teaching students Decimal-Binary-Hex as a visual. In short, here is the project, need some help on coding. I do not wish to use additional chips for coding/decoding as I think it will be easier for me to use additional pins rather than bringing outside IC's into play. I have 8 buttons with labels 1 through 8 that will represent decimal number. These are momentary push type. Have 4 LEDs in a row and also a CC 7 segment display with appropriate resistors. What I would like to happen is thus:
Press button '1' and the LSD single LED lights up, as well as the appropriate segments of the 7-segment to make the 1
Press 2 and similar... all the way to press button '7' and the 3 right LED's all light up as well as the 7 shape on digital 7seg.
"8' will make the leftmost LED light and show an '8' and so-on. I will expand on this to go up 8 LED's and (2) 7-segments.
Using a Mega 2560 so I have plenty of pins available. If needed I also have LM4 something chips that I can put to drive LEDs from digital output if necessary. If someone could please get me started as how to assign the pins to input and output state and how to make the buttons operate multiple outputs that would be of great assistance. Thanks!!!

That's not a particularly challenging project. Assumng common cathode 7 segment displays, You could use 7 pins for your annodes, then 1 pin for the cathodes of each digit. Another 4 pins for the binary LEDS.

For your button inputs you could use a resistor ladder arrangement to give you 5 buttons on just one analogue pin.

The coding for it would be quite simple. Due to persistence of vision, you can illuminate one digit, then turn off it's cathodes, present the data for the next digit then turn on it's cathodes, then turn this digit off and etc... If you swap digits say, once every 50th of a second, to the human eye it looks like all three digits are illuminated at the same time. (it's common practice).

Writing the code to achieve it would be a doddle.