Yes, all solutions are homogeneous mixtures, but not all homogeneous mixtures are solutions. If we refer to the definition of a solution, we are told that all the solutes (or things being dissolved) must be completely dissolved in the solvent with no excess.

Is a solution a mixture?

In chemistry a solution is actually a type of mixture. A solution is a mixture that is the same or uniform throughout. Think of the example of salt water. This is also called a "homogenous mixture." A mixture that is not a solution is not uniform throughout.

Are all solutions mixture are all mixtures solutions?

Solution is a homogeneous mixture of solute and solvent. The solute distributes uniformly in the solvent to form a solution. Thus, all solutions are mixtures but all mixtures are not solutions.