for example when two objects undergo a collision and no external force is acting on them. Will both kinetic energy and momentum be conserved or just momentum?
The momentum will be conserved in all cases when you consider the whole interacting system.
The total energy of the colliding bodies will also be conserved. But kinetic energy will only be conserved if the collision is elastic. It's not conserved as usually the collisions are inelastic where a bit of energy ends up as sound waves. If the bodies rub against each other during collision a bit of the energy will be also be lost as heat and a bit of energy will turn into heat if the bodies deform. The energy can be converted into other forms as well depending upon the situation.