Nema 17 with transistors
The schematic that you provided does not provide any level of functionality for driving the stepper motor. For your level of understanding of electronics you should instead get a stepper motor driver module that can connect to your motor and properly control it from your Arduino. 

There are several businesses that provide modules like this for experimenters. The ones that come to my mind are Adafruit and Sparkfun which you can find via a simple web search. The part number ROB-12779 EasyDriver from Sparkfun is a module that comes with all kinds of helpful documentation to get you started including a Schematic, Hookup Guide, Datasheet for the used A3967 chip and available Arduino code samples.

If you look up that A3967 chip data sheet and take a look at its Functional Block Diagram you can start to get a sense of what it takes to properly drive the coils on a bipolar stepper motor. For the relatively low current applications (i.e. less than 1A coil current) you are much better off using an integrated stepper motor driver chip than you are trying to build two H-Bridges all with discrete transistors and diodes.