How to make my own sequence of numbers and letters in a seven segment display circuit?
I'm pretty sure the layout of my circuit is alright.


I'd suggest NO, you have absolutely no clue if your circuit is functional. There is so much wrong with it it's hard to know where to start, so I'll go left to right in your schematic.


Read the datasheet for the 74147. It is a priority encoder, it derives the highest value signal from the inputs.


-In your case only the condition of input 9 has any relevance. You won't get a series of outputs as you think.
-You don't ground outputs you don't need ...you have grounded output "D"
-You have no pullup in the input switch. It would still work but is bad practice, get used to defining the states.  
U3A, you have an output grounded. 
U4, You have an output grounded.   


Only signals A,B have a chance of working since input C is grounded.  

The display LEDs appear to be simply 7 individual LEDS.     


Seven segment displays come as either Common Anode or Common Cathode. It would be a benefit to you to consider this configuration when working out your solution. What you have simply does not work. 





   

As an example of what does not work in your display section, you have Y0, Y1, Y2 and Y6 output resistors all connected to ground. You also have Y7 connected to ground. 

Back to the drawing board and try again. I'd suggest the almost any tutorial on using an LED seven segment display would be worth reading.

To start the learning process off I:


I suggest you start with a 74LS04 and a single LED. Understand what the output circuit looks like and understand what logic HI and Lo mean.





Look into using this invertor to drive a single LED ON/OFF with a series resistor to limit the LED current. Decide whether you want the LED ON for the output when Hi or Lo and discover the changes required to support this.
Try to use the 74LS138 to drive 7 LEDs. Activate the inputs A,B,C and see if you can get more than one LED to light (hint: you can't) with this decoder.
Have a look at the 74LS47 and see whether you can figure out how to drive the LEDs (hint: this is no open Collector drive so there are circuit restrictions).
Once you can drive 7 LEDs based on a BCD input, you are partly there. Now you need to consider how you would generate the BCD inputs required based on a single push button input.