I have sourced this sensor from a Taylor kitchen scale and would like to know what hardware I need to begin taking analog inputs from it. I’ve done some looking around and I’m thinking the h bridge included in the arduino Uno starter kit would be of some help.
Your confusion is understandable. There are many kinds of "bridges" in electronics, and they are used for many different things. There are Wheatstone bridges, diode bridges, h-bridges, and, of course, Jeff Bridges
The amplifier which amplifies load cell signal and feeds the main controller on the board might not be HX711.
Also pins like SCK and DT (found on HX711) will be internally connected to the main controller that resides on the circuit and won't be easily accessible since it is part of a complete packaged product.
Without knowing the amplifier, its' communication protocol with the main controller on the board, the lines which need to be tapped to hack data, it seems difficult to use the given circuitry with Arduino Uno.
If the above circuitry is very common then maybe you could find it's schematic online and then try to hack it.
Using a HX711 module seems the better solution rather than hacking the circuit you've posted.