Trying to work with 74HC595 manually
Anything that involves clocking- to use it with a mechanical switch usually requires debouncing and rise/fall times within some reasonable range.
The 74HC595 does not have Schmitt trigger inputs so you would need at least something like a 74HC14 externally (plus an R-C circuit) to debounce the inputs.
As well, CMOS chips require all the inputs to be tied to a valid logic level. Check out the datasheet.
There is a nSRCLR input which will reset the shift register bits to 0 when brought low. That's probably what you call MR. You would tie that high to use it normally.
The shift register outputs are only latched on the rising edge of RCLK. That input could do without debouncing since multiple edges will just latch the same value more than once.
SRCLK needs debouncing or you may get multiple bits shifted in each time you push the button.

You will also want to tie nOE low to enable the output.
So if you wire 3 pushbutton switches with one or two 74HC14 debounce stages you can operate the '595 'manually'. Press (or not) one button to set the 0/1 input, press/release a second to give a rising edge on SRCLK, repeat 8x to load the shift register, then press the third to latch the shift register value into the output latch.