First of all, it is "adsorption". Now, if you really want to see the adsorption rate or anything related to this, go for UV-Vis spectroscopy. Then perform Xrd and EDX..
UV-vis is the best method to study adsorption of dyes onto adsorbent in terms of adsorption kinetics, edges and isotherm. Other techniques are less reliable in this case. Good luck.
Well, it seems like you are interested in the adsorption of dyes onto adsorbent to be visibly understood. For this, make a comparison of the structural or morphological analyses of fresh and spent adsorbents. Your spent adsorbent can show some different pattern upon adsorption. moreover for analyses, please follow the instruments as mentioned earlier before.
UV-VIS is good for studying the adsorption process. But if you want to study the adsorbent only, then IR is an easy and direct option. This paper may help: