how can we calculate the speed of an object using the formula of Einstein $E=mc^2$?
In $E = mc^2$, $E$ is the total energy of the object, including chemical, kinetic, thermal, zero point energy, and doubtless a few others. (Zero point energy - NOT Syndrome's gloves, but the energy that something has when it's static at a temperature of 0 K.)
So, if you know the rest mass of the object, $m_0$ - what its mass would be if it currently had zero velocity, but had the same energy in all other forms - and its current mass, $m$, then you could calculate its kinetic energy as $KE = mc^2 - m_0c^2$. If the estimate $KE = \frac 12 mv^2$ indicates that $v \ll c$, then you can use the Newtonian kinetic energy formula to solve for $v$. If, however, $v$ is a significant fraction of $c$, then you will need to read up on Special Relativity - or just click the link below to HyperPhysics and try one situation at a time.
Referred to HyperPhysics.
In the comment thread you asked what is the correct equation to calculate the speed. It is impossible to answer without knowing which quantities you already know. Without considering that, it could be any equation that involves speed.
Sorry for posting this comment as an “answer”, the app won’t let me comment directly.