Lipid membranes form the outer covering of all biological cells. Embedded on the lipid membrane are numerous proteins that can diffuse on its surface due to its fluid nature. The proteins can also interact with each other through elastic and entropic forces that have their origin in the membrane's resistance to bending deformations. These interactions can be attractive or repulsive, and they likely play a role in self-assembly of proteins on the surface of the membrane to form scaffolds for exo- and endo-cytosis and also viruses. Thus, it is crucial to understand these elastic and entropic forces in detail and how they affect self-assembly of inclusions on the surface of membranes. Although most analyses of these phenomena utilize various simulation techniques, we use a semi-analytical method based on Gaussian integrals to compute the elastic and entropic interactions of inclusions. Once we have determined the interaction forces between inclusions, we use Langevin dynamics to study how they diffuse under the influence of these interaction forces. We focus first on two inclusions and cast their self-assembly as a first passage time problem. We show that an analytical treatment of the first passage time problem starting from a Fokker-Planck equation leads to a partial differential equation that can be solved numerically, and gives results which are in excellent agreement with the first passage time estimated from Langevin dynamics simulations. We are also able to account for hydrodynamic interactions between inclusions and show that they speed up the self-assembly. Finally, we use these insights to study how interaction forces influence the self-assembly of more than two inclusions. Our methods provide a different view of self-assembly that could be utilized for developing more advanced and efficient computational techniques.