This paper presents a combined analytical-numerical study for the Stokes flow caused by a rigid particle of revolution translating axisymmetrically in a viscous fluid. The fluid, which may be a slightly rarefied gas, is allowed to slip at the surface of the particle. A singularity method based on the principle of distribution of a set of Sampson spherical singularities along the axis of revolution within a prolate particle or on the fundamental plane within an oblate particle is used to find the general solution for the fluid velocity field that satisfies the boundary condition at infinity. The slip condition on the surface of the translating particle is then satisfied by applying a boundary-collocation technique to this general solution to determine the unknown coefficients. The drag force exerted on the particle by the fluid is evaluated with good convergence behavior for various cases. For the motion of a slip sphere, our drag results agree very well with the exact solution. For the translation of a no-slip spheroid, prolate or oblate, along its axis of symmetry, the agreement between our results and the analytical solutions obtained by using spheroidal coordinates is also quite good. It is found that the normalized drag force on the translating spheroid increases monotonically with an increase in the axial-to-radial aspect ratio of the spheroid for a no-slip or slightly-slip spheroid, and decreases monotonically as the ratio increases for a well-slip spheroid. The drag force on a spheroid with intermediate values of its slip coefficient is not a monotonic function of its aspect ratio. For a spheroid with a given aspect ratio, its drag force is a monotonically decreasing function of the slip coefficient of the particle.