We have developed a new technique to monitor spontaneous adsorption of fluids by porous substrates. The method is based on an optical electronic measuring system providing millisecond resolution. The method capabilities are demonstrated with the example of the absorption of millimeter-size droplets of water and aqueous solutions of polyethylene oxide and polyacrylamide by capillaries. It is shown that polymer additives even in a small amount reduce significantly the rate of adsorption. We have introduced a generalized Lucas-Washburn equation to account for the fluid elasticity. This equation is shown to explain the observed kinetics quantitatively without invoking adjustable parameters. We have derived a modified Bosanquet equation for the initial velocity of penetration, which accounts for the fluid elasticity. This simple formula gives a reasonable estimate of the rate of absorption of small droplets. We report visualization experiments on absorption of water and polymer solutions by sugar cubes as an example of porous substrates. Although the kinetics of droplet adsorption by porous substrates is similar to the kinetics of droplet adsorption by capillaries, the interpretation of experimental data is more complex and requires a plausible hydrodynamic model for lateral spreading in pores.