Implementing Push button as ON/OFF Toggle
This is a very common requirement for many logic solutions.
In TTL logic you'd use a 74LS74 or 74HC74, but you can't beat a CMOS solution such as shown below based on the CD4013 

I'll leave you work through the logic as a challenge, but note that U1a is used as an RS flip/flip and U1b used as a Dtype Toggle with reset.







simulate this circuit &ndash; Schematic created using CircuitLab
*** I have not shown power supply caps here or the supply to the VCC/Gnd to the IC.

It's worth noting that a CD4013 will cost you > $0.50  plus discrete components and you can get an ATTiny10 for $0.33.
As an exercise using the CD4013 (or 74xx74) on a breadboard may be valid, but it makes no sense when designing product. You'd learn much more using a small MCU.