Hi, I'm new to Arduino besides watching tutorials because I was to build 12 sets of 20 LEDS and have it all run together (turn on , off, and pulse). However, I'm not sure if I should buy the Arduino Uno or Arduino 2560? In addition, I have read that people use 74HC595 to power many LEDS but does that alter which Arduino I should buy/if I should use it? Thank you in advance!!
Look at purpose made LED drivers like, for example, the MAX7219 0r MAX7221 instead of shift registers.
You can then at least avoid having 240 current limiting resistors that way.
You can also chain these together, although maybe not with unlimited numbers of devices in one chain.
jchia:
Hi, I'm new to Arduino besides watching tutorials because I was to build 12 sets of 20 LEDs and have it all run together (turn on , off, and pulse). However, I'm not sure if I should buy the Arduino Uno or Arduino 2560?
Definitely neither. 
They both have a very inconvenient form factor. For a serious project, use a Nano (or a Pro Mini when you no longer need the connection to a PC).
jchia:
In addition, I have read that people use 74HC595 to power many LEDs but does that alter which Arduino I should buy/ if I should use it?
I imagine you have read that. 
Whether that is by any stretch of the imagination, a meaningful thing to do is another question entirely! 
74HC595s are not intended to drive LEDs. The MAX7219 as 6v6gt points out, is specifically designed to drive a matrix of 64 LEDs using only 16 connections. A "matrix" does not have to be arranged in a square, the 64 LEDs can be in any pattern you wish, even all in a long line. But 16 connections from the MAX7219 will connect them all.
The matrix modules available on eBay or Aliexpress are inexpensive (or were prior to Covid-19) and can be used to drive the matrix supplied or your own arrangement of LEDs, Four such modules will drive 256 LEDs - or three of your sets of 20 each - and require only three pins to connect to the Nano.
Note you will need to power the Nano - as with any of the "conventional" Arduinos - with a good regulated 5 V supply via the "5V" pin, as well as the matrix modules. Do not attempt to power via the "barrel jack" or "Vin".
12 sets of 20 LEDS
Use dedicated LED drivers!
you can use the quoted MAX7219 with bitbang or on HW-SPI. You can start with some premade modules for a quick start. After your tests, you can plug off the LED Modules and wire your own LEDs.
Or try the HT16K33 with I2C, for up to 128 LEDs
You need pulse/breath, blink? use the SX1509 (I2C). I2C will allow you to address 8 of these chips (each for 16 pulsable LEDs)