How exactly does heat get transferred in a medium? Molecular Collisions? Molecular friction?

Heat energy is ~ kinetic energy.  Molecules hava a set mass, so the change in their velocity (speed) is proportional to their KE (temp)

In a large amount (relatively large) of a sub.  the temp is proportional to the AVERAGE KE of all molecules (particles) , making up that substance.  The molecules are in constant motion and experience constant collisions with other molecules.  They do not al have the same speed.  Some at a certain time might be moving very slowly and some moving very fast.  === NO.  The molecules don't rub against each other creating a molecular friction.  But they collide .. elastic collisions mean that they transfer energy from one molecule to another (thus transferring heat).  The total energy in the molecules stays constant,  but some is transferred from one molec. to another molecule.  This is how heat energy is transferred .. by these collisions.

Temperature is a measure of the rms kinetic energy of the atoms and molecules of the material with that temperature.  Heat is proportional to the temperature for a given quantity and type of material.

Higher heat indicates higher energy and energy naturally transfers from the high to the low areas of energy.  If we have an iron rod, for example, and heat with a flame on one end, the kinetic energy of the atoms and molecules in the inflamed area with literally bump into the atoms and molecules abutting that hotter area.  So now the portion of rod next to the flamed end is also hot.  In other words, molecular collisions all along that rod are indicated by increasing temperature along that rod.

Of course, the transfer of kinetic energy from area to area is not perfect.  Some energy is lost, like to the air around the rod.  So  the heat away from the flamed end is not anywhere close to the same heat at the flamed end.