Newton's third law (action and reaction) and work done with a wheel

These are the steps through which i believe the bicycle accelerates from rest.

1. You push the pedal .

2.The back wheel moves to rotate (clockwise) to stop that relative motion $f_1$ acts forward , now the whole bicycle will accelerate forward due to $f1$.

3 . Now come to rear wheel . Since the whole bike along with rear wheel moves forward , the point of contact of rear whiles tries to skid and to oppose that , friction ($f_2$) at rear wheel acts backwards. This also creates a torque.

Translation Motion of bicycle : $f_1-f_2=M_{bicycle}a_{com}$

To answer your work done question :

1. Your muscle energy get converted into kinetic energy .If we include the friction between the gears . Your energy spent = $KE _{bicycle}+ Friction _{gears} $

But the point of the wheel that touches the ground is not moving relative to the ground; therefore, neither the reaction force nor the action force does any work; therefore, this force cannot be what causes the bike to accelerate from a standstill.

That is not correct. If the wheel is not slipping on the ground, then the static friction force will cause the bicycle, as well as the earth, to accelerate, though the latter would be imperceptible.

Per Newton's third law $F_R$= $F_A$=$F$.

Per Newton's second law the friction force $F$ that the earth exerts on the bicycle causes the bicycle to accelerate

$$a_{b}=frac{F}{m_b}$$

where $a_b$ is the acceleration of the bicycle and $m_b$ is the mass of the bicycle

The force $F$ that the bicycle exerts on the earth causes the earth to accelerate

$$a_{E}=frac{F}{M_E}$$

where $a_E$ is the acceleration of the earth and $M_E$ is the mass of the earth.

Clearly since $M_{E}>>>>m_b$, then $a_{E}<<<<a_b$ . The acceleration of the earth would be too small to measure, but it would theoretically exist nonetheless.

The chain of the bike definitely does work on the sprocket on the back axle (teeth and chain are moving with a force). Presumably, the axle does work on the bike frame; the frame and rider are pushed from the axle. Something tricky happens between the axle and the tire.

Ultimately the energy that causes work to be done on the bicycle comes from the conversion of chemical potential energy of the body to mechanical energy doing work pushing on the pedals. If we can ignore any dissipative forces in the chain of events from the force on the bike pedal to the torque applied to the wheel, then all of these forces can be considered internal to the bike/cyclist system. If we can exclude all external dissipative forces (air resistance, tire rolling resistance, etc.), then the only external force acting on the bike/cyclist system is the static friction force exerted by the earth, which is then responsible for the acceleration of the bike.

Hope this helps.