You need your pushbuttons to behave differently. You want one pushbutton for the data, 1 (pushed) and 0 (released) and one to move the data along, the clock.
So to enter a 1, press and hold the data pushbutton. While holding it down, press and release the clock pushbutton.
To enter a 0, release the data pushbutton, and press and release the clock pushbutton.
Your circuit diagram isn't clear. It's not obvious if wires join or cross. Also you have doubled up on transistors where one would do, such as the ones in the bottom left. Also the resolution doesn't show the component numbers. The transistor type isn't relevant so that shouldn't be shown.
Your circuit hasn't got any memory structures. It needs at least 8. The normal arrangement is to have the collector of one transistor connected to the base of a second, and the emitter of the second connected to the base of the first, and suitable pull-up resistors. That will remember 1 bit, because one transistor will be on, preventing the other from turning on.
A simple concept may not be the simplest circuit. A simple concept would be to have 8 masters and 8 slaves. When the clock button is pressed, the state of the master is transferred to the slave. When the clock button is released, the state of the slave is transferred to the master of the next bit in the shift register.